Law of sines questions

Question

1. Find the exact value of 75 degrees
2. Find the exact value of sin pie/12

*answer and explanation please *

Answer 1

I'm a little confused by the title to your problem because the Law of Sines isn't related to these questions.

For 1), we need to use the Angle Addition Identity:

sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

and, since 75 = 30 + 45, we can let x = 30° and y = 45° so:

sin(30 + 45) = sin(30)cos(45) + cos(30)sin(45)

Using the unit circle,

sin(30) = 1/2

cos(45) = √2/2

cos(30) = √3/2

sin(45) = √2/2

Plugging those in we get:

sin(75) = (1/2)(√2/2) + (√3/2)(√2/2) = √2/4 + √6/4 = (√2 + √6)/4

For 2) we need to use the Angle Subtraction Identity:

sin(x – y) = sin(x)cos(y) – cos(x)sin(y)

Since π/12 = 15°, and, since 55 = 45 - 30, we can let x = 45° and y = 30° so:

sin(45 – 30) = sin(45)cos(30) – cos(45)sin(30)

sin(15) = (√2/2)(√3/2) - (√2/2)(1/2)

sin(15) = sin(π/12) = √6/4 - √2/4 = (√6 - √2)/4