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First, I wanted to say that I saw your note that you have assisted over 7000 people with their questions. That is very generous of you and I would like to thank you for doing such a selfless thing :)

5.4.1

Okay, I got this question wrong on my homework today; my teacher is out on maternity leave and the substitute couldn't explain it very well, so I was hoping you could show me which of my possible answers are correct and why:

I am told to find the exact value of cos(60 degrees + 135 degrees)

My options are

A) (-1-sqrt3)/(2*sqrt2)

B) (-1+sqrt3)/(2*sqrt2)

C) (1+sqrt3) / (2*sqrt2)

D) (1-sqrt3) / (2*sqrt2)

I am very lost on this problem and could use your assistance. Thank you.

The cos(A+B) can be broken down into cosA*cosB - sinA*sinB.

If A = 60 and B = 135, that is cos60*cos135 - sin60*sin135.

Since cos60 = 1/2, cos135 = -root(2)/2, sin(60) = root(3)/2, and sin(135) = root(2)/2,

that expression turns out to be -root(2)/4 - root(6)/4.

That is the same as -(root(2)+root(6))/4.

A root(2) can be factored out, giving -root(2)(1+root(3))/4.

Since 4 = 2*root(2)root(2), root(2)/(2*root(2)root(2)) = 1/(2*root(2)).

Thus, the answer is –(1+root(3))/4, which is (-1-root(3))/4, or (A).

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