What Are Trigonometric Ratios of Special Angles?

The trigonometric ratios of special angles are the values of the six trigonometric ratios—sine, cosine, tangent, cosecant, secant, and cotangent—at commonly used angles such as 0°, 30°, 45°, 60°, and 90°.

These special angles are very important in mathematics because their trigonometric values can be found easily without using a calculator. Students frequently use them while solving problems related to trigonometry, right-angled triangles, geometry, heights and distances, and algebraic expressions.

In this article, we will learn how to find trigonometric ratios of special angles, understand the standard table, and practice the topic through short notes, MCQs, and a worksheet.

What Are Special Angles?

In elementary trigonometry, the angles 0°, 30°, 45°, 60°, and 90° are called special angles because their trigonometric ratios have exact and easily remembered values.

The most commonly used special angles are:

0°, 30°, 45°, 60°, 90°

The values of their six trigonometric ratios are:

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

Note: Rationalized forms may also be used:

• 1/√3 = √3/3
• 2/√3 = 2√3/3
• 1/√2 = √2/2

Six Trigonometric Ratios

For an angle θ in a right-angled triangle:

sin θ = Perpendicular / Hypotenuse

cos θ = Base / Hypotenuse

tan θ = Perpendicular / Base

The reciprocal ratios are:

cosec θ = Hypotenuse / Perpendicular

sec θ = Hypotenuse / Base

cot θ = Base / Perpendicular

These formulas help us understand where the values of trigonometric ratios come from.

Trigonometric Ratios of 30°

For 30°, the standard values are:

sin 30° = 1/2

cos 30° = √3/2

tan 30° = 1/√3

cosec 30° = 2

sec 30° = 2/√3

cot 30° = √3

For example:

sin 30° = 1/2

and

cosec 30° = 1/sin 30° = 2

Trigonometric Ratios of 45°

The angle 45° is particularly easy to remember because the two shorter sides of a 45°-45°-90° triangle are equal.

Therefore:

sin 45° = 1/√2

cos 45° = 1/√2

tan 45° = 1

The reciprocal ratios are:

cosec 45° = √2

sec 45° = √2

cot 45° = 1

Thus, the three most important values are:

sin 45° = cos 45° = 1/√2

and

tan 45° = 1

Trigonometric Ratios of 60°

For 60°:

sin 60° = √3/2

cos 60° = 1/2

tan 60° = √3

The reciprocal ratios are:

cosec 60° = 2/√3

sec 60° = 2

cot 60° = 1/√3

For example:

tan 60° = sin 60° / cos 60°

= (√3/2) / (1/2)

= √3

Trigonometric Ratios of 0°

At 0°, the standard values are:

sin 0° = 0

cos 0° = 1

tan 0° = 0

Their reciprocal ratios are:

cosec 0° = Not defined

sec 0° = 1

cot 0° = Not defined

Cosecant and cotangent are not defined at 0° because their definitions involve division by zero.

Trigonometric Ratios of 90°

At 90°, the values are:

sin 90° = 1

cos 90° = 0

tan 90° = Not defined

cosec 90° = 1

sec 90° = Not defined

cot 90° = 0

Tangent and secant are not defined at 90° because their formulas would require division by zero.

How to Find Trigonometric Ratios of Special Angles?

There are several ways to learn or derive the values of special angles.

One useful method is to use two standard triangles:

1. 45°-45°-90° triangle
2. 30°-60°-90° triangle

45°-45°-90° Triangle

Consider a right-angled isosceles triangle in which the two perpendicular sides are equal.

Suppose each shorter side is 1.

Using Pythagoras’ theorem:

Hypotenuse² = 1² + 1²

Hypotenuse² = 2

Therefore:

Hypotenuse = √2

Now:

sin 45° = 1/√2

cos 45° = 1/√2

tan 45° = 1

This gives the standard values for 45°.

30°-60°-90° Triangle

Consider an equilateral triangle with each side equal to 2. Draw a perpendicular from the top vertex to the base. This divides the equilateral triangle into two right-angled triangles.

The resulting triangle has:

• Hypotenuse = 2
• Shorter side = 1
• Longer side = √3

For the 30° angle:

sin 30° = 1/2

cos 30° = √3/2

tan 30° = 1/√3

For the 60° angle:

sin 60° = √3/2

cos 60° = 1/2

tan 60° = √3

The remaining three ratios can be obtained using their reciprocal relationships.

Important Reciprocal Relationships

Students should remember these relationships:

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

There are also quotient relationships:

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

These relationships make it easier to calculate a missing ratio when another ratio is known.

Easy Way to Remember sin Values

For the angles:

0°, 30°, 45°, 60°, 90°

the sine values can be remembered as:

√0/2, √1/2, √2/2, √3/2, √4/2

Therefore:

sin 0° = √0/2 = 0

sin 30° = √1/2 = 1/2

sin 45° = √2/2

sin 60° = √3/2

sin 90° = √4/2 = 1

For cosine, the same pattern is reversed:

cos 0° = √4/2 = 1

cos 30° = √3/2

cos 45° = √2/2

cos 60° = √1/2 = 1/2

cos 90° = √0/2 = 0

This is a convenient method for remembering the sine and cosine values.

Relationship Between Sine and Cosine

For the special angles, notice that the values of cosine appear in reverse order compared with sine.

For example:

sin 30° = cos 60° = 1/2

and

sin 60° = cos 30° = √3/2

Also:

sin 45° = cos 45° = √2/2

In general:

sin θ = cos(90° − θ)

and

cos θ = sin(90° − θ)

This relationship is useful when solving trigonometric problems.

Important Examples

Example 1: Find sin 60°.

From the standard table:

sin 60° = √3/2

Example 2: Find cos 30°.

We know:

cos 30° = √3/2

Therefore, the answer is:

√3/2

Example 3: Find tan 45°.

Using the standard values:

tan 45° = 1

Example 4: Find cosec 30°.

We know:

cosec θ = 1/sin θ

Therefore:

cosec 30° = 1/(1/2) = 2

Example 5: Find sec 60°.

We know:

sec θ = 1/cos θ

Therefore:

sec 60° = 1/(1/2) = 2

Example 6: Find cot 45°.

We know:

cot θ = 1/tan θ

Therefore:

cot 45° = 1/1 = 1

Why Are Special Angles Important?

The trigonometric ratios of special angles are important because they allow students to solve many problems without a calculator.

They are commonly used in:

• Trigonometric identities
• Right-angled triangles
• Geometry
• Heights and distances
• Algebraic problems
• Coordinate geometry
• Mathematical proofs
• Board examination questions

Students should try to memorize the standard table and understand how the values are derived.

Common Mistakes Students Should Avoid

1. Mixing up sine and cosine values

Remember that:

sin 30° = 1/2

but

cos 30° = √3/2

2. Forgetting reciprocal relationships

Remember:

cosec = 1/sin

sec = 1/cos

cot = 1/tan

3. Confusing tan 30° and tan 60°

tan 30° = 1/√3

while:

tan 60° = √3

4. Ignoring undefined values

Some ratios are not defined:

tan 90° = Not defined

sec 90° = Not defined

cosec 0° = Not defined

cot 0° = Not defined

Trigonometric Ratios of Special Angles: Quick Revision Table

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

For most school-level questions, learning this table accurately is extremely useful.

Short Notes: Trigonometric Ratios of Special Angles

Special angles: 0°, 30°, 45°, 60°, 90°.

Basic ratios:

• sin θ = Perpendicular/Hypotenuse
• cos θ = Base/Hypotenuse
• tan θ = Perpendicular/Base

Reciprocal ratios:

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

Important values:

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

Important identities:

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

sin θ = cos(90° − θ)

cos θ = sin(90° − θ)

MCQs: Trigonometric Ratios of Special Angles

1. What is sin 30°?

A. 0
B. 1/2
C. √2/2
D. √3/2

Answer: B. 1/2

2. What is cos 60°?

A. 1
B. √3/2
C. 1/2
D. 0

Answer: C. 1/2

3. What is tan 45°?

A. 0
B. 1/√3
C. 1
D. √3

Answer: C. 1

4. What is sin 90°?

A. 0
B. 1/2
C. √3/2
D. 1

Answer: D. 1

5. What is cos 0°?

A. 0
B. 1
C. 1/2
D. √3/2

Answer: B. 1

6. What is tan 60°?

A. 1/√3
B. 1
C. √2
D. √3

Answer: D. √3

7. What is sin 45°?

A. 1
B. 1/2
C. √2/2
D. √3/2

Answer: C. √2/2

8. What is cosec 30°?

A. 1
B. √2
C. 2
D. √3

Answer: C. 2

9. What is sec 60°?

A. 1
B. 2
C. √2
D. √3

Answer: B. 2

10. Which ratio is not defined at 90°?

A. sin 90°
B. cos 90°
C. tan 90°
D. cot 90°

Answer: C. tan 90°

11. What is cot 45°?

A. 0
B. 1/√3
C. 1
D. √3

Answer: C. 1

12. What is cos 30°?

A. 1/2
B. √2/2
C. √3/2
D. 1

Answer: C. √3/2

13. Which of the following is equal to sin 60°?

A. cos 60°
B. cos 30°
C. sin 30°
D. tan 45°

Answer: B. cos 30°

14. What is tan 30°?

A. √3
B. 1
C. 1/√3
D. √2

Answer: C. 1/√3

15. Which triangle is useful for deriving the values of 30° and 60°?

A. 30°-30°-120°
B. 30°-60°-90°
C. 45°-45°-45°
D. 60°-60°-90°

Answer: B. 30°-60°-90°

Worksheet / Assignment

Topic: Trigonometric Ratios of Special Angles

Part A: Fill in the blanks

1. sin 0° = ________
2. sin 30° = ________
3. cos 60° = ________
4. tan 45° = ________
5. sin 90° = ________
6. cos 0° = ________
7. tan 60° = ________
8. cosec 30° = ________
9. sec 60° = ________
10. cot 45° = ________

Part B: Find the following

1. sin 60°
2. cos 30°
3. tan 30°
4. tan 60°
5. sin 45°
6. cos 45°
7. cosec 60°
8. sec 30°
9. cot 30°
10. sec 0°

Part C: Complete the table

θ	sin θ	cos θ	tan θ
0°	______	______	______
30°	______	______	______
45°	______	______	______
60°	______	______	______
90°	______	______	______

Part D: Conceptual Questions

1. What are special angles?
2. Write the six trigonometric ratios.
3. Why is tan 90° not defined?
4. Why is cosec 0° not defined?
5. Explain how the values of 30° and 60° can be obtained from a 30°-60°-90° triangle.
6. Write the reciprocal relationship between sine and cosecant.
7. Write the relationship between tangent, sine, and cosine.
8. What is the value of sin 30° + cos 60°?
9. What is the value of tan 45° + cot 45°?
10. Verify that sin 60° = cos 30°.

Answer Key

Part A

1. 0
2. 1/2
3. 1/2
4. 1
5. 1
6. 1
7. √3
8. 2
9. 2
10. 1

Part B

1. √3/2
2. √3/2
3. 1/√3
4. √3
5. √2/2
6. √2/2
7. 2/√3
8. 2/√3
9. √3
10. 1

Part D

8. 1
9. 2
10. sin 60° = √3/2 and cos 30° = √3/2, so they are equal.

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Conclusion

The trigonometric ratios of special angles are an essential part of elementary trigonometry. The angles 0°, 30°, 45°, 60°, and 90° have standard values that students should learn and understand.

The most important values to remember are those of sin, cos, and tan, because the values of cosec, sec, and cot can then be obtained using reciprocal relationships.

Regular practice of the standard table, examples, MCQs, and worksheets can help students become more confident in solving trigonometry questions.

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