Applications of Trigonometric Ratios: How to Apply Trigonometric Ratios with Examples

The applications of trigonometric ratios allow us to find an unknown side or angle of a right-angled triangle when some other information is known.

Trigonometry is an important branch of mathematics that deals with the relationship between the angles and sides of triangles. The three basic trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).

In this article, we will learn how to apply trigonometric ratios, when to use sin, cos, and tan, and how to solve practical problems step by step.

These notes are prepared in a simple student-friendly style by Nisar Math Academy to help students understand the topic easily.

What Are Trigonometric Ratios?

In a right-angled triangle, trigonometric ratios establish relationships between the sides of the triangle and one of its acute angles.

The three main trigonometric ratios are:

Sine:$$\sin\theta=\frac{\text{Perpendicular}}{\text{Hypotenuse}}$$

Cosine:$$\cos\theta=\frac{\text{Base}}{\text{Hypotenuse}}$$

Tangent:$$\tan\theta=\frac{\text{Perpendicular}}{\text{Base}}$$

Here, θ represents an acute angle of the right-angled triangle.

Names of the Sides of a Right-Angled Triangle

Before learning how to use trigonometric ratios, it is important to understand the three sides of a right-angled triangle.

1. Hypotenuse

The side opposite the right angle is called the hypotenuse.

It is always the longest side of a right-angled triangle.

2. Perpendicular

The side opposite the angle θ is called the perpendicular.

It is also called the opposite side.

3. Base

The side adjacent to angle θ, other than the hypotenuse, is called the base.

The names of perpendicular and base depend on the angle being considered, while the hypotenuse remains opposite the right angle.

How to Apply Trigonometric Ratios

The basic method for applying trigonometric ratios is very simple.

Step 1: Identify the given information

First, carefully read the question and identify:

• The known angle
• The known side
• The unknown side or angle

Step 2: Identify the sides

For the given angle, determine which side is:

• Hypotenuse
• Perpendicular
• Base

Step 3: Select the appropriate ratio

Choose sin, cos, or tan according to the sides involved.

Use:$$\sin\theta=\frac{P}{H}$$

when perpendicular and hypotenuse are involved.

Use:$$\cos\theta=\frac{B}{H}$$

when base and hypotenuse are involved.

Use:$$\tan\theta=\frac{P}{B}$$

when perpendicular and base are involved.

Step 4: Substitute the values

Put the known values into the appropriate formula.

Step 5: Solve the equation

Simplify the equation to find the required side or angle.

How to Remember Sin, Cos and Tan

A common way to remember the three ratios is:

SOH – CAH – TOA

Where:

SOH$$\sin\theta=\frac{O}{H}$$

CAH$$\cos\theta=\frac{A}{H}$$

TOA$$\tan\theta=\frac{O}{A}$$

Here:

• O = Opposite
• A = Adjacent
• H = Hypotenuse

In terms of perpendicular and base:$$\sin\theta=\frac{P}{H}$$$$\cos\theta=\frac{B}{H}$$$$\tan\theta=\frac{P}{B}$$

Example 1: Finding the Perpendicular

Suppose a right-angled triangle has:$$\theta=30^\circ$$

and hypotenuse:$$H=10\text{ cm}$$

Find the perpendicular.

Solution

We know:$$\sin\theta=\frac{P}{H}$$

Substitute the values:$$\sin30^\circ=\frac{P}{10}$$

Since:$$\sin30^\circ=\frac12$$

Therefore:$$\frac12=\frac{P}{10}$$

So:$$P=5\text{ cm}$$

Answer:$$\boxed{P=5\text{ cm}}$$

Example 2: Finding the Base

Suppose:$$\theta=60^\circ$$

and the hypotenuse is 12 cm. Find the base.

Solution

Since base and hypotenuse are involved, use cosine.$$\cos\theta=\frac{B}{H}$$

Therefore:$$\cos60^\circ=\frac{B}{12}$$

We know:$$\cos60^\circ=\frac12$$

Thus:$$\frac12=\frac{B}{12}$$

Therefore:$$B=6\text{ cm}$$

Answer:$$\boxed{B=6\text{ cm}}$$

Example 3: Finding the Perpendicular Using Tangent

Suppose:$$\theta=45^\circ$$

and the base is 8 cm. Find the perpendicular.

Solution

Since perpendicular and base are involved, use tangent.$$\tan\theta=\frac{P}{B}$$

Substitute:$$\tan45^\circ=\frac{P}{8}$$

Since:$$\tan45^\circ=1$$

Therefore:$$1=\frac{P}{8}$$

Hence:$$P=8\text{ cm}$$

Answer:$$\boxed{P=8\text{ cm}}$$

Finding an Unknown Angle Using Trigonometric Ratios

Trigonometric ratios can also be used to find an unknown angle.

For example, suppose:$$P=5\text{ cm}$$

and:$$H=10\text{ cm}$$

Find θ.

Solution

Use sine:$$\sin\theta=\frac{P}{H}$$

Therefore:$$\sin\theta=\frac{5}{10}$$$$\sin\theta=\frac12$$

We know:$$\sin30^\circ=\frac12$$

Therefore:$$\boxed{\theta=30^\circ}$$

When the ratio does not correspond to a familiar angle, a calculator can be used with the inverse trigonometric function.

For example:$$\theta=\sin^{-1}(x)$$

Similarly:$$\theta=\cos^{-1}(x)$$

or$$\theta=\tan^{-1}(x)$$

depending on the ratio used.

Applications of Trigonometric Ratios in Real Life

The applications of trigonometric ratios are not limited to textbook questions. They are also used in many practical situations.

1. Finding the Height of a Building

If the distance from a building and the angle of elevation are known, trigonometric ratios can help calculate the height of the building.

2. Finding the Height of a Tree

The height of a tree can be estimated by measuring the distance from the tree and the angle of elevation to its top.

3. Finding the Length of a Ladder

If a ladder makes a particular angle with the ground and its length or distance from the wall is known, trigonometric ratios can help find the missing measurement.

4. Finding the Distance Across a River

Trigonometric relationships can be used in surveying to calculate distances that are difficult to measure directly.

5. Surveying and Construction

Engineers and surveyors use trigonometry to calculate heights, distances, slopes, and angles.

6. Navigation

Trigonometric methods are used in navigation to determine directions and distances.

7. Architecture and Engineering

Architects and engineers use trigonometric relationships when designing structures involving angles, slopes, and dimensions.

Example of a Practical Application

A student stands 20 metres away from a tree. The angle of elevation of the top of the tree is $45^\circ$. Find the height of the tree, ignoring the student’s eye height.

Solution

Let the height of the tree be $h$.

Here:$$B=20\text{ m}$$

and:$$\theta=45^\circ$$

We need to find the perpendicular, so use tangent:$$\tan\theta=\frac{P}{B}$$

Therefore:$$\tan45^\circ=\frac{h}{20}$$

Since:$$\tan45^\circ=1$$

we get:$$1=\frac{h}{20}$$

Thus:$$h=20\text{ m}$$

Answer:$$\boxed{20\text{ m}}$$

Which Trigonometric Ratio Should You Use?

A useful way to decide which ratio to use is to look at the sides involved.

Given/Required Sides	Ratio
Perpendicular and Hypotenuse	Sine
Base and Hypotenuse	Cosine
Perpendicular and Base	Tangent

Remember:$$\sin\theta=\frac{P}{H}$$$$\cos\theta=\frac{B}{H}$$$$\tan\theta=\frac{P}{B}$$

Common Mistakes Students Should Avoid

While learning how to use trigonometric ratios, students often make a few common mistakes.

Mistake 1: Choosing the Wrong Ratio

Always identify the two sides involved before selecting sin, cos, or tan.

Mistake 2: Incorrectly Identifying the Hypotenuse

The hypotenuse is always opposite the right angle and is the longest side.

Mistake 3: Using the Wrong Angle

Make sure the angle used in the ratio is the angle given in the question.

Mistake 4: Calculator in the Wrong Mode

When working with angles such as $30^\circ$, $45^\circ$, and $60^\circ$, make sure the calculator is set to degree mode when the problem is expressed in degrees.

Mistake 5: Rounding Too Early

If a calculator is required, keep sufficient decimal places during calculations and round the final answer appropriately.

Short Notes: Applications of Trigonometric Ratios

Trigonometry:
The branch of mathematics that studies relationships between angles and sides of triangles.

Sine:$$\sin\theta=\frac{P}{H}$$

Cosine:$$\cos\theta=\frac{B}{H}$$

Tangent:$$\tan\theta=\frac{P}{B}$$

SOH-CAH-TOA:

• SOH → Sine = Opposite/Hypotenuse
• CAH → Cosine = Adjacent/Hypotenuse
• TOA → Tangent = Opposite/Adjacent

Applications include:

• Finding heights
• Finding distances
• Finding unknown angles
• Surveying
• Construction
• Engineering
• Navigation
• Measuring inaccessible distances

Important: First identify the known and unknown sides, then choose the appropriate trigonometric ratio.

MCQs: Applications of Trigonometric Ratios

1. Which trigonometric ratio is equal to perpendicular divided by hypotenuse?

A. Cosine
B. Tangent
C. Sine
D. Cotangent

Answer: C. Sine

2. Which ratio is represented by base divided by hypotenuse?

A. Sin θ
B. Cos θ
C. Tan θ
D. Cot θ

Answer: B. Cos θ

3. Which ratio is equal to perpendicular divided by base?

A. Sin θ
B. Cos θ
C. Tan θ
D. Sec θ

Answer: C. Tan θ

4. What does SOH represent?

A. Sin = Opposite/Hypotenuse
B. Sin = Adjacent/Hypotenuse
C. Cos = Opposite/Hypotenuse
D. Tan = Opposite/Adjacent

Answer: A. Sin = Opposite/Hypotenuse

5. What does CAH represent?

A. Cos = Opposite/Hypotenuse
B. Cos = Adjacent/Hypotenuse
C. Tan = Adjacent/Hypotenuse
D. Sin = Adjacent/Hypotenuse

Answer: B. Cos = Adjacent/Hypotenuse

6. What does TOA represent?

A. Tan = Opposite/Adjacent
B. Tan = Adjacent/Opposite
C. Sin = Opposite/Adjacent
D. Cos = Opposite/Adjacent

Answer: A. Tan = Opposite/Adjacent

7. If the perpendicular and hypotenuse are involved, which ratio should normally be used?

A. Sin
B. Cos
C. Tan
D. None

Answer: A. Sin

8. If the base and hypotenuse are involved, which ratio should normally be used?

A. Sin
B. Cos
C. Tan
D. Cot

Answer: B. Cos

9. If the perpendicular and base are involved, which ratio should normally be used?

A. Sin
B. Cos
C. Tan
D. Sec

Answer: C. Tan

10. The hypotenuse of a right-angled triangle is:

A. The shortest side
B. The side opposite the right angle
C. The side adjacent to every angle
D. Always equal to the base

Answer: B. The side opposite the right angle

11. What is the value of $\sin30^\circ$?

A. 0
B. 1
C. 1/2
D. √3/2

Answer: C. 1/2

12. What is the value of $\cos60^\circ$?

A. 1/2
B. 1
C. √3/2
D. 0

Answer: A. 1/2

13. What is the value of $\tan45^\circ$?

A. 0
B. 1
C. √3
D. 1/2

Answer: B. 1

14. Trigonometric ratios are particularly useful for solving:

A. Right-triangle problems
B. Only arithmetic problems
C. Only algebraic equations
D. Only statistics problems

Answer: A. Right-triangle problems

15. Which of the following is a practical application of trigonometry?

A. Finding the height of a building
B. Alphabetical ordering
C. Counting letters
D. Writing paragraphs

Answer: A. Finding the height of a building

Worksheet / Assignment

Part A: Fill in the Blanks

1. \sin\theta=\frac{_____}{_____}
2. \cos\theta=\frac{_____}{_____}
3. \tan\theta=\frac{_____}{_____}
4. The longest side of a right-angled triangle is called the __________.
5. SOH stands for __________.
6. CAH stands for __________.
7. TOA stands for __________.
8. The hypotenuse is opposite the __________ angle.

Part B: Choose the Correct Ratio

Identify whether sin, cos, or tan should be used.

1. Perpendicular and hypotenuse are known.
2. Base and hypotenuse are known.
3. Perpendicular and base are known.
4. You need to find perpendicular when base and angle are known.
5. You need to find base when hypotenuse and angle are known.

Part C: Solve the Following

Question 1

A right-angled triangle has an angle of $30^\circ$ and hypotenuse of 14 cm. Find the perpendicular.

Question 2

A right-angled triangle has an angle of $60^\circ$ and hypotenuse of 20 cm. Find the base.

Question 3

A right-angled triangle has an angle of $45^\circ$ and base of 12 cm. Find the perpendicular.

Question 4

If:$$\sin\theta=\frac{1}{2}$$

find θ for an acute angle.

Question 5

If:$$\tan\theta=1$$

find θ for an acute angle.

Part D: Word Problems

Question 6

A person stands 15 m away from a tree. The angle of elevation of the top of the tree is $45^\circ$. Find the height of the tree, ignoring eye height.

Question 7

A ladder is placed against a wall. The ladder makes an angle of $60^\circ$ with the ground. If the ladder is 10 m long, find the height reached by the ladder on the wall.

Question 8

Explain any three real-life applications of trigonometric ratios.

Answer Key

Part A

1. Perpendicular, Hypotenuse
2. Base, Hypotenuse
3. Perpendicular, Base
4. Hypotenuse
5. Sin = Opposite/Hypotenuse
6. Cos = Adjacent/Hypotenuse
7. Tan = Opposite/Adjacent
8. Right

Part B

1. Sin
2. Cos
3. Tan
4. Tan
5. Cos

Part C

1. 7 cm
2. 10 cm
3. 12 cm
4. $30^\circ$
5. $45^\circ$

Part D

6. 15 m
7. $5\sqrt3$ m ≈ 8.66 m
8. Any three relevant applications, such as surveying, construction, finding heights, finding distances, engineering, or navigation.

Conclusion

The applications of trigonometric ratios provide a practical way to find unknown sides and angles in right-angled triangles. The most important step is to correctly identify the perpendicular, base, and hypotenuse according to the given angle.

Remember the three fundamental formulas:$$\boxed{\sin\theta=\frac{P}{H}}$$$$\boxed{\cos\theta=\frac{B}{H}}$$$$\boxed{\tan\theta=\frac{P}{B}}$$

Once these relationships are understood, students can solve many mathematical and real-life problems involving heights, distances, angles, slopes, surveying, construction, and engineering.

For more online math academy notes, recorded lectures, assessments, lesson plans, and Class 9 Mathematics learning resources, visit Nisar Math Academy at www.nisarmathacademy.com. Students can access selected free resources, while membership provides access to the full course material.

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