find the value of sin 75 and cos 75

To find the value of sin 75 and cos 75, we can use the sum of angles identities.

For sin 75, we can break it down into sin (45 + 30).

The formula for sin(a + b) is sin a cos b + cos a sin b.

So, sin 75 = sin (45 + 30) = sin 45 cos 30 + cos 45 sin 30.

We know that sin 45 = cos 45 = √2/2, sin 30 = 1/2, and cos 30 = √3/2.

Substituting these values in, we get sin 75 = (√2/2 * √3/2) + (√2/2 * 1/2) = √6/4 + √2/4 = (√6 + √2) / 4.

For cos 75, we can also break it down into cos (45 + 30).

The formula for cos(a + b) is cos a cos b - sin a sin b.

So, cos 75 = cos (45 + 30) = cos 45 cos 30 - sin 45 sin 30.

Substituting the known values in, we get cos 75 = (√2/2 * √3/2) - (√2/2 * 1/2) = √6/4 - √2/4 = (√6 - √2) / 4.

So, sin 75 = (√6 + √2) / 4 and cos 75 = (√6 - √2) / 4.