Retrieving exact value using Compound angle

[aeromat]

Homework Statement

Homework Equations

The Attempt at a Solution

I'm stuck. I don't know what to do when the angle cannot be the sum or difference of two special angles (like 45,60,30). I tried taking a look at other topics, but there wasn't a clear solution for me to follow.

 

[Bohrok]

For b, try 30° - 45° = -15°

For c, 11$\pi$/12 = $\pi$ - $\pi$/12, or 175° = 180° - 15° = 180° + 30° - 45°

 

[aeromat]

Ok, you showed me more than one way to get to the required angle. Would all the various ways end up giving me the same answer?

 

[HallsofIvy]

How about you try each way and see?

 

[aeromat]

If I do 45-60, and 30-45 I get (I used tan(a-b), where 'a' is smaller and 'b' is the larger value)

1 - root of 3
------------ <-- divided by
1 + root of 3

correct answer is:

-2 + root of 3.. <-- no denominator

How do they get that?

 

[eumyang]

If I do 45-60, and 30-45 I get (I used tan(a-b), where 'a' is smaller and 'b' is the larger value)

1 - root of 3
------------ <-- divided by
1 + root of 3

correct answer is:

-2 + root of 3.. <-- no denominator

How do they get that?

Multiply the numerator and denominator by the conjugate of the denominator:
$$\frac{1 - \sqrt{3}}{1 + \sqrt{3}} \cdot \frac{1 - \sqrt{3}}{1 - \sqrt{3}}=...$$
It's not considered simplified if you have a rational expression with a radical in the denominator, so that's why we do this.