Angle Difference Formula

The angle difference formulas are used to find the trigonometric ratios of some specific non-standard angles by writing them as the difference of two standard angles. These are also known as angle difference identities. Along with angle difference formulas, we have angle sum formulas as well.

What is Angle Difference Formula?

We already know the values of trigonometric functions of standard angles from the trigonometric table. i.e., we can directly tell the exact values of sin 90°, cos 45°, cosec 30°, etc using the trigonometric table. But what are the values of sin 75°, cos 15°, tan 105°, etc? These can be found using the angle difference formulas and angle sum formulas. Here are the angle difference formulas.

• sin (A - B) = sin A cos B - cos 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)

You can see the angle difference and angle sum formulas below.

Let us see the applications of the angle difference formulas in the following section.

Examples Using Angle Difference Formula

Example 1: Find the exact value of sin 15°.

Solution:

To find: The value of sin 15°.

Using the one of the angle difference formulas (of sin),

sin (A - B) = sin A cos B - cos A sin B

Substitute A = 45° and B = 30° in the above formula,

sin (45° - 30°) = sin 45° cos 30° - cos 45° sin 30°

sin 15° = (1 / √2) (√3 / 2) - ( 1/ √2) (1 / 2) 

= √3 / 2√2 - 1 / 2√2

= (√3 - 1) / 2√2

Answer: sin 15° = (√3 - 1) / 2√2

Example 2: Find the exact value of cot 15°.

Solution:

To find: The value of cot 15°.

Using one of the angle difference formulas (of tan),

tan (A - B) = (tan A - tan B) / (1 + tan A tan B)

Substitute A = 45° and B = 30° in the above formula,

tan (45° - 30°) = (tan 45° - tan 30°) / (1 + tan 45° tan30°)

tan 15° = (1 - (√3 / 3)) / ( 1 + 1 × (√3 / 3))

= ( (3 - √3) / 3) / ( (3 + √3) /3 )

= (3 - √3) / (3 + √3)

But we want to find cot 15°. We know that cot and tan are reciprocals of each other.

cot 15° = 1 / (tan 15°)

Applying the angle diference formula, tan (A - B) = (tan A - tan B) / (1 + tan A tan B)

tan 15° = (tan 45° - tan 30°)/(1+ tan 45° tan 30°)

cot 15° = (1+ tan 45° tan 30°)/(tan 45° - tan 30°)

= (1+ 1 . 1/√3)/ (1- 1/√3)

= (1+ 1/√3)/ (1- 1/√3)

= (√3 + 1)/  (√3 - 1)

Rationalizing the denominator,

cot 15° =  (1 + √3) / (1 - √3) ×  (1 + √3) / (1 + √3)

= (1+ √3) ² / (3-1)

= (1 + 3 + 2 √3)/2

= (4+ 2 √3)/2

= 2 + √3.

Answer: cot 15° = 2 + √3.