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Standard Trigonometric Ratios

Reading Time: 45 mins

Solved Examples: 20

Practice Questions: 5

What are Standard Trigonometric Ratios?

Standard Trigonometric Ratios are the exact values of the six trigonometric ratios sin θ, cos θ, tan θ, cosec θ, sec θ, and cot θ for the commonly used angles 0°, 30°, 45°, 60°, and 90°. These values are fixed and can be obtained without using a calculator. They form the foundation of trigonometry and are widely used to solve mathematical problems involving right-angled triangles and angles. Understanding these standard values makes calculations faster and helps in solving more advanced trigonometric problems efficiently.

Standard Angles

Standard angles are the five special angles for which the exact values of the trigonometric ratios are known.
The standard angles are:

0°
30°
45°
60°
90°

Standard Trigonometric Ratios Table

θ	0°	30°	45°	60°	90°
sin θ	0	½	√2⁄2	√3⁄2	1
cos θ	1	√3⁄2	√2⁄2	½	0
tan θ	0	1⁄√3	1	√3	Not Defined
cosec θ	Not Defined	2	√2	2⁄√3	1
sec θ	1	2⁄√3	√2	2	Not Defined
cot θ	Not Defined	√3	1	1⁄√3	0

Relationship Between sin θ and cos θ

The values of sin θ increase as the angle increases from 0° to 90°, whereas the values of cos θ decrease over the same interval.
The following relationship always holds:
sin θ = cos (90° − θ)
Similarly, cos θ = sin (90° − θ)
Examples:

sin 30° = cos 60° = ½
sin 45° = cos 45° = √2⁄2
sin 60° = cos 30° = √3⁄2

Reciprocal Relationships of Standard Ratios

The reciprocal trigonometric ratios are obtained by taking the reciprocal of the corresponding primary ratio.
Formulae:
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ
For example,

If sin 30° = ½, then cosec 30° = 2
If cos 60° = ½, then sec 60° = 2
If tan 45° = 1, then cot 45° = 1

Whenever the denominator becomes 0, the reciprocal ratio is Not Defined.

cosec 0° is Not Defined
sec 90° is Not Defined
tan 90° is Not Defined

How to Remember the Standard Values

The values of sin θ can be remembered using the following pattern:
sin θ = √n⁄2
where n = 0, 1, 2, 3, 4 for the angles 0°, 30°, 45°, 60°, 90°, respectively. Thus,

sin 0° = √0⁄2 = 0
sin 30° = √1⁄2 = ½
sin 45° = √2⁄2
sin 60° = √3⁄2
sin 90° = √4⁄2 = 1

The values of cos θ are obtained by reading the same sequence in reverse order.
The remaining four trigonometric ratios can then be found using:

tan θ = sin θ ÷ cos θ
cosec θ = 1 ÷ sin θ
sec θ = 1 ÷ cos θ
cot θ = 1 ÷ tan θ

This pattern provides a quick and reliable way to remember all the standard trigonometric ratios without memorising each value individually.

Summary of Standard Trigonometric Ratios

Topic	Summary
Standard Angles	The standard angles are 0°, 30°, 45°, 60° and 90°. The exact values of the trigonometric ratios at these angles are fixed.
Primary Ratios	The three primary trigonometric ratios are sin θ, cos θ and tan θ. The remaining three ratios are their reciprocals.
Complementary Relationship	sin θ = cos (90° − θ) and cos θ = sin (90° − θ).
Reciprocal Relationships	cosec θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ.
Undefined Values	tan 90°, cosec 0° and sec 90° are Not Defined because their denominators become zero.
Range of Values	As θ increases from 0° to 90°, sin θ increases from 0 to 1, while cos θ decreases from 1 to 0.
Remembering sin θ	For 0°, 30°, 45°, 60° and 90°, sin θ follows the pattern √0⁄2, √1⁄2, √2⁄2, √3⁄2 and √4⁄2.
Remembering cos θ	The values of cos θ are obtained by reading the values of sin θ in the reverse order.
Finding tan θ	tan θ = sin θ ÷ cos θ, provided cos θ ≠ 0.
Finding the Remaining Ratios	Once sin θ and cos θ are known, tan θ, cosec θ, sec θ and cot θ can be determined using quotient and reciprocal relationships.

Solved Examples

Question 1: Find the value of sin 30° + cos 60°

sin 30° = ½
cos 60° = ½
Therefore,
sin 30° + cos 60°
= ½ + ½
= 1

Question 2: Find the value of tan 45° + sec 60° − cosec 30°

tan 45° = 1
sec 60° = 2
cosec 30° = 2
Therefore,
1 + 2 − 2
= 1

Question 3: Find the value of (sin 60° × cos 30°) + (sin 45°)²

sin 60° = √3/2
cos 30° = √3/2
sin 45° = √2/2
Therefore,
= (√3/2 × √3/2) + (√2/2)²
= 3/4 + 2/4
= 5/4

Question 4: Find the value of (tan 60° × cot 30°) − (sec 0° × cos 45°)

tan 60° = √3
cot 30° = √3
sec 0° = 1
cos 45° = √2/2
Therefore,
= 3 − √2/2

Question 5: Find the value of [(sin 60° + cos 60°)²] − (tan 30° × cot 60°)

sin 60° + cos 60°
√3/2 + ½
= (√3 + 1)/2
Squaring,
= (√3 + 1)²/4
= (3 + 1 + 2√3)/4
= 1 + √3/2
Also,
tan 30° × cot 60°
= (1/√3) × (1/√3)
= 1/3
Therefore,
= 1 + √3/2 − 1/3
= 2/3 + √3/2

Question 6: Find the value of (sin 30° × cosec 30°) + (cos 60° × sec 60°) + (tan 45° × cot 45°)

sin 30° × cosec 30°
= ½ × 2 = 1
cos 60° × sec 60°
= ½ × 2 = 1
tan 45° × cot 45°
= 1 × 1 = 1
Therefore,
= 1 + 1 + 1
= 3

Question 7: Find the value of (tan² 60° + cot² 30°) ÷ sec² 60°

tan 60° = √3
cot 30° = √3
sec 60° = 2
Therefore,
= (3 + 3)/4
= 6/4
= 3/2

Question 8: Find the value of (1/sec 60°) + (1/cosec 30°)

1/sec 60°
= cos 60°
= ½
1/cosec 30°
= sin 30°
= ½
Therefore,
= ½ + ½
= 1

Question 9: Find the value of (1/sin 45°) + (1/cos 60°) − (1/tan 45°)

1/sin 45°
= cosec 45°
= √2
1/cos 60°
= sec 60°
= 2
1/tan 45°
= cot 45°
= 1
Therefore,
= √2 + 2 − 1
= √2 + 1

Question 10: Find the value of (sec² 60° − tan² 60°)

cosec 30° = 2
cot 30° = √3
Therefore,
= 2² − (√3)²
= 4 − 3
= 1

Question 11: Find the value of [(sec 60° × cosec 30°) − (cot 45° × tan 45°)]

sec 60° = 2
cosec 30° = 2
cot 45° = 1
tan 45° = 1
Therefore,
= (2 × 2) − (1 × 1)
= 4 − 1
= 3

Question 12: Find the value of (tan 60° × tan 30°) + (cot 30° × cot 60°)

tan 60° = √3
tan 30° = 1/√3
cot 30° = √3
cot 60° = 1/√3
Therefore,
= 1 + 1
= 2

Question 13: Find the value of [(tan 30° + cot 60°)/(sec 30° + cosec 60°)]

tan 30° = 1/√3
cot 60° = 1/√3
Therefore,
Numerator
= 2/√3
Also,
sec 30° = 2/√3
cosec 60° = 2/√3
Denominator
= 4/√3
Hence,
= (2/√3)/(4/√3)
= 1/2

Question 14: Arrange the following in ascending order sin 30°, cos 60°, tan 45°, cos 45°

sin 30° = ½
cos 60° = ½
cos 45° = √2/2
tan 45° = 1
Since
½ < √2/2 < 1
Therefore,
sin 30° = cos 60° < cos 45° < tan 45°

Question 15: Arrange the following in descending order tan 30°, tan 45°, tan 60°

tan 30° = 1/√3
tan 45° = 1
tan 60° = √3
Since
√3 > 1 > 1/√3
Therefore,
tan 60° > tan 45° > tan 30°

Question 16: Arrange the following in ascending order cos 0°, sec 60°, cosec 45°, tan 60°

cos 0° = 1
cosec 45° = √2
tan 60° = √3
sec 60° = 2
Since
1 < √2 < √3 < 2
Therefore,
cos 0° < cosec 45° < tan 60° < sec 60°

Question 17: Which is greater? (sin 60° + cos 30°) or (tan 45° + cos 0°)

sin 60° + cos 30°
= √3/2 + √3/2
= √3
tan 45° + cos 0°
= 1 + 1
= 2
Since
2 > √3
Therefore,
tan 45° + cos 0° is greater.

Question 18: Arrange the following in descending order cosec 60°, sec 45°, tan 45°, cos 30°

cosec 60° = 2/√3
sec 45° = √2
tan 45° = 1
cos 30° = √3/2
Since
√2 > 2/√3 > 1 > √3/2
Therefore,
sec 45° > cosec 60° > tan 45° > cos 30°

Question 19: Evaluate (sin² 45° + cos² 45°) + (tan² 60° − sec² 60°)

sin² 45° + cos² 45°
= ½ + ½
= 1
tan² 60°
= 3
sec² 60°
= 4
Therefore,
= 1 + (3 − 4)
= 1 − 1
= 0

Question 20: If x = sin 30° + cos 60° + tan 45° and y = sec 60° − cosec 30° + cot 45° find the value of x² + y².

x
= ½ + ½ + 1
= 2
y
= 2 − 2 + 1
= 1
Therefore,
x² + y²
= 2² + 1²
= 4 + 1
= 5

Common Mistakes

Memorising Standard Values Incorrectly:
Students often confuse the values of sin θ and cos θ for standard angles. Remember that the values of sin θ increase from 0° to 90°, while the values of cos θ decrease over the same range.
Formula:
sin 0° = 0, sin 90° = 1
cos 0° = 1, cos 90° = 0
Confusing Complementary Angle Relationships:
Students sometimes interchange the complementary relationships incorrectly. Remember that the trigonometric ratio changes to its complementary ratio when the angle is replaced by (90° − θ).
Formulae:
sin θ = cos (90° − θ)
cos θ = sin (90° − θ)
tan θ = cot (90° − θ)
Using Reciprocal Ratios Incorrectly:
Students often confuse reciprocal pairs. Remember that cosec θ is the reciprocal of sin θ, sec θ is the reciprocal of cos θ, and cot θ is the reciprocal of tan θ.
Formulae:
cosec θ = 1 ÷ sin θ
sec θ = 1 ÷ cos θ
cot θ = 1 ÷ tan θ
Ignoring Undefined Values:
Students sometimes calculate values where the denominator becomes zero and write an incorrect answer. Remember that division by zero is not defined.
Examples:
tan 90° is Not Defined.
sec 90° is Not Defined.
cosec 0° is Not Defined.
Making Errors While Simplifying Expressions:
Students often make mistakes while adding, subtracting or multiplying trigonometric values involving surds. Always simplify carefully, rationalise denominators where required, and verify that the final answer is in its simplest form.

Practice Questions

Question 1: Arrange the following trigonometric ratios in ascending order tan 30°, cos 45°, sin 60°, tan 60°

Question 2: Find the value of (sec 60° × cos 30°) + (cosec 30° × sin 60°)

Question 3: Find the value of [(1 − cos 60°) ÷ (1 + cos 60°)] + [(1 − sin 30°) ÷ (1 + sin 30°)]

Question 4: Find the value of (tan 60° × cot 60°) + (sec 45° × cos 45°) + (cosec 90° × sin 90°)

Question 5: Simplify (sec 60° − tan 45°)² + (cosec 30° − cot 45°)²