Find the Value of sin 135°

Find the value of sin 135°. 

Value of Sin 135°

Angle 135° can be broken into :

• Addition of 90° and 45°

• Subtraction of 180° and 45°

1. sin 135° can be written as sin ( 90° + 45° )

As sin 135° lies between angle 90 °🩂and 180°, it will be in the second quadrant  where :

Sine will be positive. 

• When using the identity or the trigonometric functions

Identity: sin (a + b) = sin (a). cos (b) + cos (a). sin (b) 

Hence, 

         = sin (90°) cos (45°) + cos(90°) sin (45°) 

         = (1 × 1/√2)   +  (0 × 1/√2)

         = 1/√2

When turning into a simpler form, 

( rationalize the denominator by √2 )

         = √2/2

• When using the direct method 

Sin 135° lies in the second quadrant and is positive 

Sin 135° = sin (90° + 45° )

(Note: sin (90° + x )= cos x )

         = cos 45° (in the first quadrant )

( Note: the cosine is positive in the first quadrant )

         = 1/√2

2. Sin 135° can be written as sin (180° – 45°)

          Hence, it lies in the second quadrant 

• When using the identity for calculation or using the trigonometric functions

Identity: sin (a – b) = sin (a). cos(b).  – cos (a). sin(b)

= sin 180°. cos 45° – cos 180°. sin 45°

= 1/√2

When rationalising the denominator 

= √2/2

•  When a direct format is used 

           = sin (180° – 45°)

         ( Note: sine (180° – x )= sin x) 

           = sin 45° ( in the first quadrant ) 

           ( Note: sine is positive in the first quadrant )

           = 1/√2

The result of sin 135° can be shown in different forms 

• 1/√2

• Simpler form after rationalize :

√2/2

•  Decimal form: = 0.70710678……. (exact value)

In the above case, the value of sin 135° is found positive because the angle lies in the second quadrant and sine is always positive in the second quadrant.