What Are Trigonometric Ratios? Definition, Formulas, Table, Examples & Notes

What Are Trigonometric Ratios?

Trigonometric ratios are ratios of the sides of a right-angled triangle. They are used to find unknown sides and angles of a triangle and are an important part of mathematics, especially in geometry and trigonometry.

For a given acute angle in a right-angled triangle, the three sides are named:

• Hypotenuse: The side opposite the right angle. It is always the longest side.
• Perpendicular (Opposite): The side opposite the angle under consideration.
• Base (Adjacent): The side next to the angle under consideration, other than the hypotenuse.

There are six commonly used trigonometric ratios:

1. Sine (sin)
2. Cosine (cos)
3. Tangent (tan)
4. Cosecant (cosec)
5. Secant (sec)
6. Cotangent (cot)

Trigonometric Ratios Formula

Suppose ABC is a right-angled triangle with ∠B = 90° and we are considering angle A.

Then:

Hypotenuse = AC

Perpendicular/Opposite = BC

Base/Adjacent = AB

The trigonometric ratios formulae are:

1. Sine

$$\sin A=\frac{\text{Perpendicular}}{\text{Hypotenuse}}$$

2. Cosine

$$\cos A=\frac{\text{Base}}{\text{Hypotenuse}}$$

3. Tangent

$$\tan A=\frac{\text{Perpendicular}}{\text{Base}}$$

4. Cosecant

Cosecant is the reciprocal of sine:$$\cosec A=\frac{\text{Hypotenuse}}{\text{Perpendicular}}$$

5. Secant

Secant is the reciprocal of cosine:$$\sec A=\frac{\text{Hypotenuse}}{\text{Base}}$$

6. Cotangent

Cotangent is the reciprocal of tangent:$$\cot A=\frac{\text{Base}}{\text{Perpendicular}}$$

Trigonometric Ratios Table

Ratio	Formula
sin A	Perpendicular / Hypotenuse
cos A	Base / Hypotenuse
tan A	Perpendicular / Base
cosec A	Hypotenuse / Perpendicular
sec A	Hypotenuse / Base
cot A	Base / Perpendicular

A useful way to remember the first three ratios is:

SOH – CAH – TOA

• SOH: Sin = Opposite / Hypotenuse
• CAH: Cos = Adjacent / Hypotenuse
• TOA: Tan = Opposite / Adjacent

Example of Trigonometric Ratios

Suppose a right-angled triangle has:

• Perpendicular = 3 cm
• Base = 4 cm
• Hypotenuse = 5 cm

For angle A:$$\sin A=\frac{3}{5}$$$$\cos A=\frac{4}{5}$$$$\tan A=\frac{3}{4}$$

The reciprocal ratios are:$$\cosec A=\frac{5}{3}$$$$\sec A=\frac{5}{4}$$$$\cot A=\frac{4}{3}$$

Therefore, the six trigonometric ratios can be calculated directly from the three sides of a right-angled triangle.

Relationship Between Trigonometric Ratios

The trigonometric ratios are related to one another.$$\tan A=\frac{\sin A}{\cos A}$$

Also,$$\cot A=\frac{\cos A}{\sin A}$$

The reciprocal relationships are:$$\sin A=\frac{1}{\cosec A}$$$$\cos A=\frac{1}{\sec A}$$$$\tan A=\frac{1}{\cot A}$$

These relationships are useful when simplifying trigonometric expressions.

Trigonometric Ratios of Important Angles

Students should learn the values of trigonometric ratios for some common angles.

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	Not defined

The values of cosec, sec and cot can be obtained by taking the reciprocals of sin, cos and tan respectively.

How to Learn Trigonometric Ratios

Many students find trigonometric ratios difficult at first, but they become much easier with regular practice.

Step 1: Understand the Three Sides

First learn to identify:

Perpendicular, Base and Hypotenuse.

Remember that the hypotenuse is always opposite the right angle.

Step 2: Learn SOH-CAH-TOA

Use:

SOH → Sin = Opposite/Hypotenuse

CAH → Cos = Adjacent/Hypotenuse

TOA → Tan = Opposite/Adjacent

Step 3: Understand Reciprocal Ratios

Remember:

• cosec = reciprocal of sin
• sec = reciprocal of cos
• cot = reciprocal of tan

Step 4: Practice Right-Angled Triangles

Start with simple triangles whose sides are known. Calculate all six ratios.

Step 5: Memorize Important Angle Values

Practice the values of 0°, 30°, 45°, 60° and 90°.

Step 6: Solve Different Questions

Regular practice is the best way to become confident with trigonometric ratios.

Trigonometric Ratios Calculator

A trigonometric ratios calculator can be useful for checking calculations involving sin, cos and tan. Most scientific calculators have buttons for:

sin, cos and tan

For example, to calculate:$$\sin 30^\circ$$

make sure the calculator is in degree (DEG) mode and enter 30 using the sin function.

You should still understand the formulas before relying on a calculator, especially when preparing for school examinations.

Important Points to Remember

1. Trigonometric ratios are based on the sides of a right-angled triangle.
2. The hypotenuse is always opposite the right angle.
3. The hypotenuse is the longest side of a right-angled triangle.
4. There are six basic trigonometric ratios.
5. Sin, cos and tan are the three most commonly used ratios.
6. Cosec, sec and cot are reciprocal ratios.
7. The formulas depend on the angle being considered.
8. SOH-CAH-TOA is useful for remembering sin, cos and tan.
9. Always identify the correct opposite and adjacent sides before applying a formula.
10. A calculator should normally be set to degree mode when working with angles such as 30°, 45° and 60°.

Short Notes on Trigonometric Ratios

Definition: Trigonometric ratios are ratios between the sides of a right-angled triangle with respect to an acute angle.

Six Trigonometric Ratios:$$\sin\theta=\frac{P}{H}$$$$\cos\theta=\frac{B}{H}$$$$\tan\theta=\frac{P}{B}$$$$\cosec\theta=\frac{H}{P}$$$$\sec\theta=\frac{H}{B}$$$$\cot\theta=\frac{B}{P}$$

Where:

P = Perpendicular/Opposite

B = Base/Adjacent

H = Hypotenuse

Memory Trick:

SOH – CAH – TOA

Sin = Opposite/Hypotenuse

Cos = Adjacent/Hypotenuse

Tan = Opposite/Adjacent

Reciprocal Ratios:$$\cosec\theta=\frac{1}{\sin\theta}$$$$\sec\theta=\frac{1}{\cos\theta}$$$$\cot\theta=\frac{1}{\tan\theta}$$

MCQs on Trigonometric Ratios

1. What are trigonometric ratios based on?

A. Circle only
B. Right-angled triangle
C. Square
D. Rectangle

Answer: B. Right-angled triangle

2. Which side is opposite the right angle?

A. Base
B. Perpendicular
C. Hypotenuse
D. Adjacent

Answer: C. Hypotenuse

3. What is the formula for sin θ?

A. Base/Hypotenuse
B. Perpendicular/Base
C. Perpendicular/Hypotenuse
D. Hypotenuse/Base

Answer: C. Perpendicular/Hypotenuse

4. What is the formula for cos θ?

A. Base/Hypotenuse
B. Hypotenuse/Base
C. Perpendicular/Base
D. Base/Perpendicular

Answer: A. Base/Hypotenuse

5. What is the formula for tan θ?

A. Base/Hypotenuse
B. Perpendicular/Base
C. Hypotenuse/Perpendicular
D. Hypotenuse/Base

Answer: B. Perpendicular/Base

6. Which ratio is the reciprocal of sin θ?

A. sec θ
B. cot θ
C. cosec θ
D. tan θ

Answer: C. cosec θ

7. Which ratio is the reciprocal of cos θ?

A. sec θ
B. cosec θ
C. cot θ
D. tan θ

Answer: A. sec θ

8. Which ratio is the reciprocal of tan θ?

A. sin θ
B. cos θ
C. cot θ
D. sec θ

Answer: C. cot θ

9. What is sin 30°?

A. 0
B. 1/2
C. √3/2
D. 1

Answer: B. 1/2

10. What is tan 45°?

A. 0
B. 1/2
C. 1
D. √3

Answer: C. 1

11. What is cos 60°?

A. 1
B. √3/2
C. 1/2
D. 0

Answer: C. 1/2

12. Which memory rule helps remember sin, cos and tan?

A. ABC
B. SOH-CAH-TOA
C. PEMDAS
D. BODMAS

Answer: B. SOH-CAH-TOA

Worksheet / Assignment

Part A: Fill in the Blanks

1. The longest side of a right-angled triangle is called the __________.
2. The ratio of perpendicular to hypotenuse is called __________.
3. The ratio of base to hypotenuse is called __________.
4. The ratio of perpendicular to base is called __________.
5. The reciprocal of sin θ is __________.
6. The reciprocal of cos θ is __________.
7. The reciprocal of tan θ is __________.
8. The formula for tan θ is __________.
9. The value of sin 30° is __________.
10. The value of tan 45° is __________.

Part B: Write the Formula

Write the formula for:

1. sin θ
2. cos θ
3. tan θ
4. cosec θ
5. sec θ
6. cot θ

Part C: Calculate the Ratios

A right-angled triangle has:

Perpendicular = 5 cm

Base = 12 cm

Hypotenuse = 13 cm

Find:

1. sin θ
2. cos θ
3. tan θ
4. cosec θ
5. sec θ
6. cot θ

Part D: Find the Values

Find the following:

1. sin 30°
2. cos 60°
3. tan 45°
4. sin 90°
5. cos 0°
6. tan 0°

Part E: Conceptual Questions

1. What are trigonometric ratios?
2. Why is the hypotenuse always the longest side?
3. Explain the difference between base and perpendicular.
4. Write and explain SOH-CAH-TOA.
5. What are reciprocal trigonometric ratios?
6. Where are trigonometric ratios used in mathematics?

Conclusion

Trigonometric ratios provide a simple way to relate the sides and angles of a right-angled triangle. The six basic ratios—sin, cos, tan, cosec, sec and cot—are fundamental concepts that students need to understand before moving on to more advanced topics in trigonometry.

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