Question

If A = 45°, verify that `sin A = sqrt((1 - cos 2A)/2)`.

Answer 1

Given: A = 45°

Step-wise calculation:

1. Left side: sin A = sin 45°

= `1/sqrt(2)` 

= `sqrt(2)/2`

2. Right side: `sqrt((1 - cos 2A)/2`.

Since 2A = 90°, cos 90° = 0.

So, `sqrt((1 - cos 2A)/2`

= `sqrt((1 - 0)/2`

= `sqrt(1/2)`

= `1/sqrt(2)`

= `sqrt(2)/2`

3. Because A = 45° is in the first quadrant sin A is positive, so taking the positive square root is correct.

`sin A = sqrt((1 - cos 2A)/2` holds for A = 45° both sides = `sqrt(2)/2`.