problem solving with trigonometric ratios answer key

TRIGONOMETRIC RATIOS EXAMPLES AND SOLUTIONS

Example problems and solutions given in this section will be much useful for the students who would like to practice problems on trigonometric ratios.

Before look at the example problems, if you would like to know the basic stuff on trigonometric ratios,

please click here

Example 1 :

Compare the sine, the cosine, and the tangent ratios for  ∠ A in each triangle below.

By the SSS Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion, which implies that the trigonometric ratios for  ∠ A in each triangle are the same.

Trigonometric ratios are frequently expressed as decimal approximations.

Example 2 :

Find the sine, the cosine, and the tangent of the indicated angle.

a.   ∠S          b.   ∠R

Solution (a) :

The length of the hypotenuse is 13. For  ∠ S, the length of the opposite side is 5, and the length of the adjacent side is 12.

sin S  =  opp./hyp.  =  5/13   ≈  0.3846

cos S  =  adj./hyp.  =  12/13   ≈  0.9231

tan S  =  opp./adj.  =  5/12   ≈  0.4167

Solution (b) :

The length of the hypotenuse is 13. For  ∠ R, the length of the opposite side is 12, and the length of the adjacent side is 5.

sin R  =  opp./hyp.  =  12/13   ≈  0.9231

cos R  =  adj./hyp.  =  5/13   ≈  0.3846

tan R  =  opp./adj.  =  12/5   ≈  2.4

Example 3 :

Find the sine, the cosine, and the tangent of 45 °. 

Begin by sketching a 45°-45°-90° triangle. Because all such triangles are similar, we can make calculations simple by choosing 1 as the length of each leg. From the Pythagorean Theorem, it follows that the length of the hypotenuse is  √2.

sin 45 °   =  opp./hyp.  =  1/ √2  =   √2/2    ≈  0.7071

cos 45 °   =  adj./hyp.  =  1/ √2  =   √2/2   ≈  0.7071

tan 45 °   =  opp./adj.  =  1/1  =  1

Example 4 :

Find the sine, the cosine, and the tangent of 30 °. 

Begin by sketching a 30°-60°-90° triangle. To make the calculations simple, we can choose 1 as the length of the shorter leg. From  30°-60°-90° Triangle theorem ,  it follows that the length of the longer leg is  √3 and the hypotenuse is 2.

sin 30 °   =  opp./hyp.  =  1/ 2  =   0.5

cos 30 °   =  adj./hyp.  =  √3/2   ≈  0.8660   

tan 30 °   =  opp./adj.  =  1/ √3  =   √3/3   ≈  0.5774

Example 5 :

Jacob is measuring the height of a Sitka spruce tree in North Carolina. He stands 45 feet from the base of the tree. He measures the angle of elevation from a point on the ground to the top of the tree to be 59°. How can he estimate the height of the tree ? 

To estimate the height of the tree, Jacob can write a trigonometric ratio that involves the height h and the known length of 45 feet.

In the above right triangle, for the angle 59°, h is opposite side and the side has length 45 ft is adjacent side. 

The trigonometric ratio that involves opposite side and adjacent side is tangent. 

Write ratio : 

tan 59°  =  opp./adj. 

Substitute. 

tan 59°  =  h/45

Multiply each side by 45. 

45  ⋅ tan 59 °  =  h

Use calculator or table to find the value of tan 59 ° and substitute. 

45  ⋅ 1.6643  =   h

74.9   ≈  h

So, the tree is about 75 feet tall. 

Example 6 :

The escalator at the University Metro Rail Station near  University of Miami in Coral Gables, Florida  rises  76 feet at a 30° angle as shown in the diagram below. Find the distance d a person travels on the escalator stairs. 

To find the distance d a person travels on the escalator stairs, we  can write a trigonometric ratio that involves the side has length d and the known length of 76 feet.

In the above right triangle, for the angle 30°, d is hypotenuse and the side has length 76 feet is opposite side. 

The trigonometric ratio that involves opposite side and hypotenuse is sin. 

sin 30°  =  opp./hyp.

sin 30°  =  76/d

Multiply each side by d. 

d  ⋅ sin 30 °  =  76

Substitute 0.5 for sin  30 °.

d  ⋅ 0.5  =  76   

Divide each side by 0.5

d  =   76/0.5

d  =  152

So, the person travels 152 feet on the escalator stairs. 

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Solve Problems Using Trigonometric Ratios

Trigonometric Ratio Worksheets

How to Use Trigonometric Ratios to Finding Missing Sides of a Triangle - Trigonometric functions are vastly used in real-life situations. One of the most common uses of these functions is to solve triangles. You can use a trigonometric function to solve a triangle in three basic steps; You need to start by choosing the appropriate trigonometric function, including sine, cosine, or tangent. This can be done through the determination of which side you know and which side you are looking for. The next step is substitution; whatever information you have, you can insert it into the functions. Once you have substituted the values within the function, it is time to solve it.

Basic Lesson

Guides students solving ratio based triangle problems. In right triangle ABC, Hypotenuse AB = 15 and angle A = 35°. Find the length of BC, to the nearest tenth.

Intermediate Lesson

Demonstrates how to missing legs and the hypotenuse based on ratios.In right triangle ABC, leg BC = 15 and leg AC = 20. Find angle A to the nearest degree. Tan A = Opposite Leg / Adjacent Leg.

Independent Practice 1

A really great activity for allowing students to understand the concept of trigonometric ratios.

Independent Practice 2

Students find the final value of missing trigonometric Ratios in assorted problems.

Homework Worksheet

Students are provided with problems to achieve the concepts of missing trigonometric ratios.

This tests the students ability to evaluate trigonometric ratios.

Homework and Quiz Answer Key

Answers for the homework and quiz.

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Trigonometric Ratios Worksheets

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. Trigonometry ratios worksheets are known for its many identities. Trigonometric ratios are defined as the ratios between edges of a right triangle. Trigonometric ratios worksheets help students understand trigonometric identities which are used for simplifying an expression.

Benefits of Trigonometric Ratios Worksheets

Trigonometry is widely used in day-to-day applications. By solving Trigonometric ratios worksheets, you'll understand that the trigonometric ratios are given by the trigonometric functions of the known angle A, where a, b, and c are lengths of the sides of the figure produced.

Real-life examples of trigonometric ratio include areas such as astronomy, geodesy, surveying, and navigation.

Download Trigonometric Ratios Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

Trigonometric Ratios

A series of free, online High School Geometry Video Lessons and solutions. In these lessons, we will learn

• how to find the sine of an angle in a right triangle,
• how to find the cosine of an angle in a right triangle,
• how to find the tangent of an angle in a right triangle,
• how to use inverse trigonometric functions to find an angle with a given trigonometric value.

Related Pages Using SOH-CAH-TOA Trigonometry Word Problems Inverse trigonometry Lessons On Trigonometry Trigonometry Worksheets

The following diagram shows the trigonometric ratios using SOHCAHTOA. Scroll down the page if you need more examples and solutions on how to use the trigonometric ratios.

Trigonometric Ratios: Sine

Right triangles have ratios to represent the angles formed by the hypotenuse and its legs. Sine ratios, along with cosine and tangent ratios, are ratios of the lengths of two sides of the triangle. Sine ratios in particular are the ratios of the length of the side opposite the angle they represent over the hypotenuse. Sine ratios are useful in trigonometry when dealing with triangles and circles.

How to define the sine ratio and identify the sine of an angle in a right triangle?

Identify the hypotenuse of a right triangle. sin θ = opposite/hypotenuse

A word problem involving the trigonometric ratio of sine to calculate the height of a pole

Example: A 55 ft wire connects a point on the ground to the top of a pole. The cable makes an angle of 60° with the ground. Find the height of the pole to the nearest foot.

Trigonometric Ratios: Cosine

Right triangles have ratios that are used to represent their base angles. Cosine ratios, along with sine and tangent ratios, are ratios of two different sides of a right triangle.Cosine ratios are specifically the ratio of the side adjacent to the represented base angle over the hypotenuse. In order to find the measure of the angle, we must understand inverse trigonometric functions.

How to use the Cosine formula (the CAH Formula) to find missing sides or angles?

Cosine θ = adjacent/hypotenuse

Example: Find the missing side or angle a) cos 30° = x/2 b) cos 37° = 4.2/x c) cos θ = 63/80

How to apply the Sine and Cosine Ratios?

Sine, cosine are trigonometric ratios for the acute angles and involve the length of a leg and the hypotenuse of a right triangle.

Angle of elevation - When looking up at an object, the angle your line of sight makes with a horizontal line is called the angle of elevation.

Angle of depression - When looking down at an object, the angle your line of sight makes with a horizontal line is called the angle of depression.

• Find sin U and sin W. Write each answer as a fraction and as a decimal rounded to four places.
• Find cos S and cos R. Write each answer as a fraction and as a decimal rounded to four places.
• You walk from one corner of a basketball court to the opposite corner. write and solve a proportion using a trigonometric ratio to approximate the distance of the walk.
• You are at the top of a roller coaster 100 feet above the ground. The angle of depression is 44°. About how far do you ride down the hill?
• A railroad crossing arm that is 20 feet long is stuck with an angle of elevation of 35°. Find the lengths of x and y.
• Use a special right triangle to find the sine and cosine of a 30° angle.

Trigonometric Ratios: Tangent

Right triangles have ratios that are used to represent their base angles. Tangent ratios, along with cosine and sine ratios, are ratios of two different sides of a right triangle. Tangent ratios are the ratio of the side opposite to the side adjacent the angle they represent. In order to find the measure of the angle itself, one must understand inverse trigonometric functions.

How to use the Tangent formula (the TOA Formula)?

Tangent θ = opposite/adjacent

Example: Find the missing side or angle a) tan 28° = x/40 b) tan 41° = 1.9/x c) tan θ = 11/8

Applications of Trigonometric Ratios (Word Problems Involving Tangent, Sine and Cosine)

• Find the area of the parallelogram.
• A 70 foot ramp rises from the first floor to the second floor of a parking garage. The ramp makes an angle with the ground. How high above the first floor is the second floor?
• You see Mr. Wandera flying a kite in the park. The kite string is 65 meters long. What angle does the string need to form with the ground so that the kite is 30 feet off of the ground?
• From the top of a 100-foot lookout tower, a forest ranger spotted a fire at a 25° angle of depression. How far was the fire from the base of the lookout tower?
• An 8 foot ladder is leaning against a wall. The ladder makes a 53° angle with the wall. How high does the ladder reach?

Inverse Trigonometric Functions

Once we understand the trigonometric functions sine, cosine, and tangent, we are ready to learn how to use inverse trigonometric functions to find the measure of the angle the function represents. Inverse trigonometric functions, found on any standard scientific or graphing calculator, are a vital part of trigonometry and will be encountered often in Calculus.

How to use inverse trigonometric functions to find an angle with a given trigonometric value and how to use inverse trigonometric functions to solve a right triangle?

Example: Use the calculator to find an angle θ in the interval [0, 90] that satisfies the equation.

• sin θ = 0.7523
• tan θ = 3.54
• Solve the given right triangle if a = 44.3 cm and b = 55.9 cm.
• Find each angle in a 3,4,5 triangle

How to use inverse trig to find a missing angle?

Inverse trig functions are used to find missing angles rather than missing sides.

Find missing Angles - Using Inverse Sine, Cosine, Tangent

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Problems on Trigonometric Ratios

Some trigonometric solutions based problems on trigonometric ratios are shown here with the step-by-step explanation.

1. If sin θ = 8/17, find other trigonometric ratios of <θ.

Let us draw a ∆ OMP in which ∠M = 90°.

Then sin θ = MP/OP = 8/17.

Let MP = 8k and OP = 17k, where k is positive.

By Pythagoras’ theorem, we get

Therefore, sin θ = MP/OP = 8k/17k = 8/17

cos θ = OM/OP = 15k/17k = 15/17

tan θ = Sin θ/Cos θ = (8/17 × 17/15) = 8/15

csc θ = 1/sin θ = 17/8

sec θ = 1/cos θ = 17/15 and

cot θ = 1/tan θ = 15/8.

2. If Cos A = 9/41, find other trigonometric ratios of ∠A.

Let us draw a ∆ ABC in which ∠B = 90°.

Then cos θ = AB/AC = 9/41.

Let AB = 9k and AC = 41k, where k is positive.

Therefore, sin A = BC/AC = 40k/41k = 40/41

cos A = AB/AC = = 9k/41k = 9/41

tan A = Sin A/Cos A = (40/41 × 41/9) = 40/9

csc A = 1/sin A = 41/40

sec A = 1/cos A = 41/9 and

cot A = 1/tan A = 9/40.

3. Show that the value of sin θ and cos θ cannot be more than 1.

We know, in a right angle triangle the hypotenuse is the longest side.

sin θ = perpendicular/hypotenuse = MP/OP < 1 since perpendicular cannot be greater than hypotenuse; sin θ cannot be more than 1.

Similarly, cos θ = base/hypotenuse = OM/OP < 1 since base cannot be greater than hypotenuse; cos θ cannot be more than 1.

4. Is that possible when A and B be acute angles, sin A = 0.3 and cos B = 0.7?

Since A and B are acute angles, 0 ≤ sin A ≤ 1 and 0 ≤ cos B ≤ 1, that means the value of sin A and cos B lies between 0 to 1. So, it is possible that sin A = 0.3 and cos B = 0.7

5. If 0° ≤ A ≤ 90° can sin A = 0.4 and cos A = 0.5 be possible?

Let us draw a ∆ ABC in which ∠B = 90° and ∠BAC = θ.

Then sin θ = BC/AC = 1/2.

Let BC = k and AC = 2k, where k is positive.

= 3√3/2 - 4 × 3√3/8

= 3√3/2 - 3√3/2

Hence, (3cos θ - 4 cos³ θ) = 0.

7. Show that sin α + cos α > 1 when 0 ° ≤ α ≤ 90°

From the right triangle MOP,

Sin α = perpendicular/ hypotenuse

Cos α = base/ hypotenuse

Now, Sin α + Cos α

= perpendicular/ hypotenuse + base/ hypotenuse

= (perpendicular + base)/hypotenuse, which is > 1, Since we know that the sum of two sides of a triangle is always greater than the third side.

8. If cos θ = 3/5, find the value of (5csc θ - 4 tan θ)/(sec θ + cot θ)

Let ∠A = θ°

Then cos θ = AB/AC = 3/5.

Let AB = 3k and AC = 5k, where k is positive.

Therefore, sec θ = 1/cos θ = 5/3

tan θ = BC/AB =4k/3k = 4/3

cot θ = 1/tan θ = 3/4 and

csc θ = AC/BC = 5k/4k = 5/4

Now (5csc θ -4 tan θ)/(sec θ + cot θ)

= (5 × 5/4 - 4 × 4/3)/(5/3 + 3/4)

= (25/4 -16/3)/(5/3 +3/4)

= 11/12 × 12/29

9. Express 1 + 2 sin A cos A as a perfect square.

1 + 2 sin A cos A

10. If sin A + cos A = 7/5 and sin A cos A =12/25, find the values of sin A and cos A.

sin A + cos A = 7/5

⇒ cos A = 7/5 - sin θ

Now from sin θ/cos θ = 12/25

We get, sin θ(7/5 - sin θ) = 12/25

or, 5 sin θ(5 sin θ - 4) - 3(5 sin θ - 4) = 0

or, (5 sin θ - 3) (5 sin θ - 4) = 0

⇒ (5 sin θ - 3) = 0 or, (5 sin θ - 4) = 0

⇒ sin θ = 3/5 or, sin θ = 4/5

When sin θ = 3/5, cos θ = 12/25 × 5/3 = 4/5

Again, when sin θ = 4/5, cos θ = 12/25 × 5/4 = 3/5

Therefore, sin θ =3/5, cos θ = 4/5

or, sin θ =4/5, cos θ = 3/5.

11. If 3 tan θ = 4, evaluate (3sin θ + 2 cos θ)/(3sin θ - 2cos θ).

Solution: Given,

3 tan θ = 4

⇒ tan θ = 4/3

(3sin θ + 2 cos θ)/(3sin θ - 2cos θ)

= (3 tan θ + 2)/(3 tan θ - 2), [dividing both numerator and denominator by cos θ]

= (3 × 4/3 + 2)/(3 × 4/3 -2), putting the value of tan θ = 4/3

12. If (sec θ + tan θ)/(sec θ - tan θ) = 209/79, find the value of θ.

Solution: (sec θ + tan θ)/(sec θ - tan θ) = 209/79

⇒ [(sec θ + tan θ) - (sec θ - tan θ)]/[(sec θ + tan θ) + (sec θ - tan θ)] = [209 – 79]/[209 + 79], (Applying componendo and dividendo)

⇒ 2 tan θ/2 sec θ =130/288

⇒ sin θ/cos θ × cos θ = 65/144

⇒ sin θ = 65/144.

13. If 5 cot θ = 3, find the value of (5 sin θ - 3 cos θ)/(4 sin θ + 3 cos θ).

Given 5 cot θ = 3

⇒ cot θ = 3/5

Now (5 sin θ - 3 cos θ)/(4 sin θ + 3 cos θ)

= (5 - 3 cot θ)/(4 sin θ + 3 cot θ), [dividing both numerator and denominator by sin θ]

= (5 - 3 × 3/5)/(4 + 3 × 3/5)

= (5 - 9/5)/(4 + 9/5)

= (16/5 × 5/29)

⇒ sin θ(sin θ - 2) - 1(sin θ - 2) = 0

⇒ (sin θ - 2)(sin θ - 1) = 0

⇒ (sin θ - 2) = 0 or, (sin θ - 1) = 0

⇒ sin θ = 2 or, sin θ = 1

So, value of sin θ can’t be greater than 1,

Therefore sin θ = 1

Basic Trigonometric Ratios  

Relations Between the Trigonometric Ratios

Reciprocal Relations of Trigonometric Ratios

Trigonometrical Identity

Problems on Trigonometric Identities

Elimination of Trigonometric Ratios  

Eliminate Theta between the equations

Problems on Eliminate Theta  

Trig Ratio Problems

Proving Trigonometric Ratios

Trig Ratios Proving Problems

Verify Trigonometric Identities  

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Primary Trig Ratios Worksheets | Sine, Cosine, Tangent

Practice these assortments of primary trigonometric ratio worksheets designed to benefit high school children with topics like finding trigonometric ratios using SOH-CAH-TOA; determine all the three primary trigonometric ratios of the right-angled triangle; find the value of sine, cosine and tangent; and more. Our free trigonometric ratios worksheets are definitely worth a try!

Primary Trigonometric Ratios Using Segments

These printable worksheets require students to use SOH-CAH-TOA to find the trigonometric ratios sin θ, cos θ and tan θ.

• Download the set

Primary Trigonometric Ratios Using Lengths

The sides of the right-angled triangle are given in these pdf worksheets. Determine the value of the given trigonometric ratio. The skill is offered in both customary and metric units.

Find sin, cos, tan: Level 1

All the three sides of a right-angled triangle are given. Find the value of sine, cosine and tangent for the specified angle.

Find sin, cos, tan: Level 2

Apply Pythagorean theorem to determine the missing length of a right-angled triangle. Find all the three primary trigonometric ratios using the information provided.

Indicated Primary Trigonometric Ratio

Each problem contains a trigonometric ratio. Determine the value of the indicated trigonometric ratio. Ex: Find sin θ if cos θ = 3/5.

Solve Using Calculator

Use calculator to determine the value of sine, cosine or tangent. Each printable high school worksheet consists of two sections, Part-A and Part-B that offers problems in degrees and radians respectively.

Evaluate Trigonometric Expressions

Evaluate the expressions in these pdf worksheets by identifying the values of the given trigonometric ratios. The angles are specified in degrees or radians.

Related Worksheets

» Pythagorean Theorem

» Trigonometry Charts

» Reciprocal Trigonometric Ratios

» Trigonometric Ratios

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Table of Contents

Last modified on August 3rd, 2023

#ezw_tco-2 .ez-toc-title{ font-size: 120%; ; ; } #ezw_tco-2 .ez-toc-widget-container ul.ez-toc-list li.active{ background-color: #ededed; } chapter outline

Trigonometric ratios.

Trigonometric ratios are the ratios of the side lengths of a right-angled triangle. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan). There are three more trigonometric ratios, cosecant (cosec), secant (sec), and cotangent (cot), that is, the inverse of sine, cosine, and tangent.

Thus, it is used to determine the ratios of any 2 sides of a right-angled triangle with respect to specific angles.

The value of these trigonometric ratios is calculated using the measure of any special acute angle θ in a right-angled triangle.

The standard formulas for the given trigonometric ratios for θ (where θ is an acute angle) are:

Sine (sin) :  It is the ratio of the opposite side (perpendicular side) to θ to the hypotenuse

∴ sin θ = Opposite side/ Hypotenuse

Cosine (cos) : It is the ratio of the adjacent side to θ to the hypotenuse

∴ cos θ = Adjacent side/ Hypotenuse

Tangent (tan) : It is the ratio of the opposite side to θ to the side adjacent to θ

∴ tan θ = Opposite side/ Adjacent side

Cosecant (cosec) : It is the multiplicative inverse of sine

∴ cosec θ = 1/sin θ = Hypotenuse /Opposite side

Secant (sec) : It is the multiplicative inverse of cosine

∴ sec θ = 1/cos θ = Hypotenuse/ Adjacent side

Cotangent (cot) : It is the multiplicative inverse of tangent

∴ cot θ = 1/tan θ = Adjacent side/Opposite side

In the given right-triangle ABC, the trigonometric ratios are shown:

Thus, with respect to ∠C, the ratios of trigonometry are:

• sine C = Side opposite to ∠C/Hypotenuse = AB/AC
• cos C = Side adjacent to ∠C/Hypotenuse = BC/AC
• tan C = Side opposite to ∠C/Side adjacent to ∠) = AB/BC = sin ∠C/cos ∠C
• cosec C= 1/sin C = Hypotenuse/ Side Opposite to ∠C = AC/AB
• sec C = 1/cos C = Hypotenuse/ Side Opposite to ∠C = AC/BC
• cot C = 1/tan C = Side adjacent to ∠C/Side opposite to ∠C = BC/AB

All the ratios written above for ∠C in all possible right triangles will be the same. The same being with ∠A

An Easy Way to Remember Trigonometric Ratios

The word sohcahtoa can be used to remember the trigonometric ratio formulas of sine, cosine, and tangent:

Trigonometric Ratios Table

The trigonometric ratios table displays the values of trigonometric ratios for the standard angles 0°, 30°, 45°, 60°, and 90° used in several trigonometric calculations.  

Trigonometric Ratios Identities

Several trigonometric ratios identities make our calculations simpler such as:

• sin 2  θ + cos 2  θ = 1
• 1 + tan 2  θ = sec 2  θ
• 1 + cot 2  θ = cosec 2  θ

There are also some variations of the above 3 identities, which are nothing but rearranging the ones given above.

Trigonometric Ratios of Complementary Angles Identities

The complement of an angle θ is (90° – θ). Accordingly, the trigonometric ratios of complementary angles are:

• sin (90°- θ) = cos θ
• cos (90°- θ) = sin θ
• cosec (90°- θ) = sec θ
• sec (90°- θ) = cosec θ
• tan (90°- θ) = cot θ
• cot (90°- θ) = tan θ

Other Trigonometric Ratios Identities

The sum, difference, and product trigonometric ratios identities are:

• sin (A + B) = sin A cos B + cos A sin B
• sin (A – B) = sin A cos B – cos A sin B
• cos (A + B) = cos A cos B – sin A sin B
• cos (A – B) = cos A cos B + sin A sin B
• tan (A + B) = (tan A + tan B)/ (1 – tan A tan B)
• tan (A – B) = (tan A – tan B)/ (1 + tan A tan B)
• cot (A + B) = (cot A cot B – 1)/(cot B – cot A)
• cot (A – B) = (cot A cot B + 1)/(cot B – cot A)
• 2 sin A⋅cos B = sin(A + B) + sin(A – B)
• 2 cos A⋅cos B = cos(A + B) + cos(A – B)
• 2 sin A⋅sin B = cos(A – B) – cos(A + B)

The half, double, and triple-angles trigonometric ratios identities

• sin 2θ = 2 sinθ cosθ
• cos 2θ = cos 2 θ – sin 2 θ = = 2 cos 2 θ – 1 = 1 – 2 sin 2 θ = (1 – tan 2 θ)/(1 + tan 2 θ)
• sec 2θ = sec 2 θ/(2-sec 2 θ)
• cosec 2θ = (sec θ. cosec θ)/2
• cot 2θ = (cot θ – tan θ)/2
• sin 3θ = 3sin θ – 4sin 3 θ
• cos 3θ = 4cos 3 θ – 3cos θ
• tan 3θ = (3tanθ – tan 3 θ)/(1 – 3tan 2 θ)

Find the value of tan θ if sin θ = 14/5 and cos θ =6/9

As we know, tan θ = sin θ/cos θ, here sin θ = 14/5 and cos θ = 6/9 = (14/5)(6/9) = 63/15

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Trigonometric Ratios

The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). In geometry, trigonometry is a branch of mathematics that deals with the sides and angles of a right-angled triangle. Therefore, trig ratios are evaluated with respect to sides and angles. 

The trigonometry ratios for a specific angle ‘θ’ is given below:

Note: Opposite side is the perpendicular side and the adjacent side is the base of the right-triangle. Also, check out trigonometric functions to learn about each of these ratios or functions in detail. Trigonometric Identities

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.   The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle .

The three sides of the right triangle are:

• Hypotenuse (the longest side)
• Perpendicular (opposite side to the angle)
• Base (Adjacent side to the angle)

Related Articles:

• Trigonometric Identities
• Trigonometric Ratios of Complementary Angles
• Trigonometric Ratios Of Standard Angles
• Trigonometry Angles
• Trigonometry Formulas
• Trigonometry Values

How to Find Trigonometric Ratios?

Consider a right-angled triangle, right-angled at B.

With respect to ∠C, the ratios of trigonometry are given as:

• sine: Sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.
• cosine: Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.
• tangent: Tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.
• cosecant: Cosecant is a multiplicative inverse of sine.
• secant: Secant is a multiplicative inverse of cosine.
• cotangent: Cotangent is the multiplicative inverse of the tangent.

The above ratios are abbreviated as sin, cos, tan, cosec, sec and tan respectively in the order they are described. So, for Δ ABC , the ratios are defined as:

sin C = ( Side opposite to ∠C)/(Hypotenuse)  = AB/AC

cos C  = ( Side adjacent to ∠C)/(Hypotenuse)  = BC/AC

tan C  = ( Side opposite to ∠C)/(Side adjacent to ∠C)  = AB/BC  = sin ∠C/cos ∠C

cosec C = 1/sin C = ( Hypotenuse)/ (Side Opposite to ∠C)  =  AC/AB

sec C  = 1/cos C = ( Hypotenuse)/ (Side Opposite to ∠C)  = AC/BC

cot C  = 1/tan C  = ( Side adjacent to ∠C)/(Side opposite to ∠C) = BC/AB

In right Δ ABC,  if ∠ A  and ∠ C  are assumed as 30°  and 60° , then there can be infinite right triangles with those specifications but all the ratios written above for ∠ C  in all of those triangles will be same. So, all the ratios for any of the acute angles (either ∠ A  or ∠ C ) will be the same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle.

Trigonometric Ratios Table

The trigonometric ratios for some specific angles such as 0 °, 30 °, 45 °, 60 ° and 90° are given below, which are commonly used in mathematical calculations.

From this table, we can find the value for the trigonometric ratios for these angles. Examples are:

• Sin 30° = ½
• Cos 90° = 0
• Tan 45° = 1 

Trigonometry Applications

Trigonometry is one of the most important branches of mathematics. Some of the applications of trigonometry are:

• Measuring the heights of towers or big mountains
• Determining the distance of the shore from the sea
• Finding the distance between two celestial bodies
• Determining the power output of solar cell panels at different inclinations
• Representing different physical quantities such as mechanical waves, electromagnetic waves, etc.

It is evident from the above examples that trigonometry has its involvement in a major part of our day-to-day life and much more. In most of the applications listed above, something was being measured and that is what trigonometry is all about.

Solved Problems

Q.1: If in a right-angled triangle ABC, right-angled at B, hypotenuse AC = 5cm, base BC = 3cm and perpendicular AB = 4cm and if ∠ACB = θ, then find tan θ, sin θ and cos θ.

Sol: Given, 

Hypotenuse, AC = 5cm 

Base, BC = 3cm 

Perpendicular, AB = 4cm

tan θ = Perpendicular/Base = 4/3

Sin θ = Perpendicular/Hypotenuse = AB/AC = ⅘

Cos θ = Base/Hypotenuse = BC/AC = ⅗

Q.2: Find the value of tan θ if sin θ = 12/5 and cos θ = ⅗.

Sol: Given, sin θ = 12/5 and cos θ = ⅗

As we know,

Tan θ = Sin θ/Cos θ

Tan θ = (12/5)/(⅗)

Tan θ = 12/3

Practice Questions

• Find the value of sin θ, if tan θ = ¾ and cos θ = ½. 
• Find tan θ if sin θ = 4/3 and cos θ = 3/2
• Find sec θ, if cos θ = 9/8
• Find cosec θ, if sin θ 16/5

Video Lesson

Trigonometric ratios of compound angles.

Frequently Asked Questions – FAQs

What are the three primary trigonometric ratios, what are the six trigonometric ratios, what is soh cah toa, what is the formula for cotangent, secant and cosecant, what is the relationship between sin, cos and tan.

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Tnks for this! Very informative!

The explanation of trigonometric ratios is given very neat and illustrative. It is easy to understand and difficult to forget.

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5. Trigonometry

5.2 Solve Applications: Sine, Cosine and Tangent Ratios.

Learning Objectives

By the end of this section it is expected that you will be able to

• Find missing side of a right triangle using sine, cosine, or tangent ratios
• Find missing angle of a right triangle using sine, cosine, or tangent ratios
• Solve applications using right angle trigonometry

Sine, Cosine, and Tangent Ratios

We know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use θ (theta) to represent it.

The hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.

In the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

Label the sides of the triangle and find the hypotenuse, opposite, and adjacent.

We labeled  the sides with a lowercase letter to match the uppercase letter of the opposite vertex.

c is hypotenuse

a is opposite

b is adjacent

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

y is hypotenuse

z is opposite

x is adjacent

Trigonometric Ratios

Trigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.

Let us refer to Figure 1 and define the three basic trigonometric ratios as:

Three Basic Trigonometric Ratios

Where θ is the measure of a reference angle measured in degrees.

Very often we use the abbreviations for sine, cosine, and tangent ratios.

Some people remember the definition of the trigonometric ratios as SOH CAH TOA.

For the given triangle find the sine, cosine and tangent ratio.

For the given triangle find the sine cosine and tangent ratio.

When calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.

For the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.

We have two possible reference angles: R an S.

Using the definitions, the trigonometric ratios for angle R are:

Using the definitions, the trigonometric ratios for angle S:

For the given triangle find the sine, cosine, and tangent ratios. If necessary round to four decimal places.

Now, let us use a scientific calculator to find the trigonometric ratios.  Can you find the sin, cos, and tan buttons on your calculator? To find the trigonometric ratios make sure your calculator is in Degree Mode.

Using a calculator find the trigonometric ratios. If necessary, round to 4 decimal places.

Make sure your calculator is in Degree Mode.

a)  Using a calculator find that sin 30° = 0.5

b)  Using a calculator find that cos 45° = 0.7071 Rounded to 4 decimal places.

c)  Using a calculator find that tan 60° = 1.7321 Rounded to 4 decimal places.

Find the trigonometric ratios. If necessary, round to 4 decimal places.

a)  sin 60°

b)  cos 30°

c)  tan 45°

a)   sin 60° = 0.8660

b)   cos 30° = 0.8660

c)   tan 45° = 1

Finding Missing Sides of a Right Triangle

In this section you will be using trigonometric ratios to solve right triangle problems. We will adapt our problem solving strategy for trigonometry applications. In addition, since those problems will involve the right triangle, it is helpful to draw it (if the drawing is not given) and label it with the given information.We will include this in the first step of the problem solving strategy for trigonometry applications.

HOW TO: Solve Trigonometry Applications

• Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.
• Identify what we are looking for.
• Label what we are looking for by choosing a variable to represent it.
• Find the required trigonometric ratio.
• Solve the ratio using good algebra techniques.
• Check the answer by substituting it back into the ratio in step 4 and by making sure it makes sense in the context of the problem.
• Answer the question with a complete sentence.

In the next few examples, having given the measure of one acute angle and the length of one side of the right triangle, we will solve the right triangle for the missing sides.

 Find the missing sides. Round  your final answer to two decimal places

 Find the missing sides. Round  your final answer to one decimal place.

Find the hypotenuse. Round your final answer to one decimal place.

Finding Missing Angles of a Right Triangle

Sometimes we have a right triangle with only the sides given. How can we find the missing angles? To find the missing angles, we use the inverse of the trigonometric ratios. The inverse buttons sin -1 , cos -1 , and tan -1 are on your scientific calculator.

Find the angles. Round your final answer to one decimal place.

a)  sin A = 0.5

b)  cos B = 0.9735

c)  tan C = 2.89358

Use your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key

a)  A = sin -1 0.5

b)  B = cos -1 0.9735

c) C = tan -1 2.89358

a)  sin X = 1

b)  cos Y = 0.375

c)  tan Z = 1.676767

In the example below we have a right triangle with two sides given. Our acute angles are missing. Let us see what the steps are to find the missing angles.

Find the missing angle X. Round your final answer to one decimal place.

Find the missing angle A. Round your final answer to one decimal place.

Find the missing angle C. Round your final answer to one decimal place.

Solving a Right Triangle

From the section before we know that any triangle has three sides and three interior angles. In a right triangle, when all six parts of the triangle are known, we say that the right triangle is solved.

Solve the right triangle. Round your final answer to one decimal place.

Since the sum of angles in any triangle is 180°, the measure of angle B can be easy calculated.

We solved the right triangle

Solve the right triangle. Round to two decimal places.

The missing angle F = 180° – 90° – 26.39° = 63.64°

Solve the right triangle. Round to one decimal place.

Solve Applications Using Trigonometric Ratios

In the previous examples we were able to find missing sides and missing angles of a right triangle. Now, let’s use the trigonometric ratios to solve real-life  problems.

Many applications of trigonometric ratios involve understanding of an angle of elevation or angle of depression.

The angle of elevation is an angle between the horizontal line (ground) and the observer’s line of sight.

The angle of depression is the angle between horizontal line (that is parallel to the ground) and the observer’s line of sight.

James is standing 31 metres away from the base of the Harbour Centre in Vancouver. He looks up to the top of the building at a 78° angle. How tall is the Harbour Centre?

Marta is standing 23 metres away from the base of the tallest apartment building in Prince George and looks at the top of the building at a 62° angle. How tall is the building?

43.3 metres

Thomas is standing at the top of the building that is 45 metres high and looks at her friend that is standing on the ground, 22 metres from the base of the building. What is the angle of depression?

Hemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend that is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?

Key Concepts

• Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label  the given parts.
• Check the answer by substituting it back into the ratio solved in step 5 and by making sure it makes sense in the context of the problem.

5.2 Exercise Set

Label the sides of the triangle.

• If the reference angle in Question 1 is B, Find the adjacent ?
• If the reference angle in Question 2 is Z, find the opposite?

Use your calculator to find the given ratios. Round to four decimal places if necessary:

For the given triangles, find the sine, cosine and tangent of the θ.

For the given triangles, find the missing side. Round it to one decimal place.

For the given triangles, find the missing sides. Round it to one decimal place.

Solve the triangles. Round to one decimal place.

• Kim stands 75 metres from the bottom of a tree and looks up at the top of the tree at a 48° angle. How tall is the tree?
• A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52°. How tall is the tree?
• A ladder that is 15 feet is leaning against a house and makes a 45° angle with the ground. How far is the base of the ladder from the house?
• Roxanne is flying a kite and has let out 100 feet of string. The angle of elevation with the ground is 38°. How high is her kite above the ground?
• Marta is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation?
• An airplane takes off from the ground at the angle of 25°. If the airplane traveled 200 kilometres, how high above the ground is it?

• y = 19.3, z = 8.2

Attribution:

This chapter has been adapted from “Solve Applications: Sine, Cosine and Tangent Ratios” in Introductory Algebra  by Izabela Mazur, which is under a CC BY 4.0 Licence . See the Adaptation Statement for more information.

Business/Technical Mathematics Copyright © 2021 by Izabela Mazur and Kim Moshenko is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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SOHCAHTOA Worksheet and Answer Key

Students will practice identifying adjacent, opposite sides (and hypotenuse) in right triangles and they will practice writing Sine Cosine Tangent (SOHCAHTOA) relationships. This sheet has model problems worked out, step by step -- as well as, 20 scaffolded questions that start out relatively easy and end with some real challenges.

Example Questions

Ii. identify sides opposite, adjacent to angles.

III. Use SOHCAHTOA

Visual Aids

Other Details

This is a 4 part worksheet:

• Part I Model Problems
• Part II Practice
• Part III Challenge Problems
• Part IV Answer Key
• Sine Cosine Calculator
• Arcsine Calculator

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