Question

If sin a = .25 and cos b = .25, which of the following is sin b/2 + cos a/2
A. 1.60
B. 1.32
C. .26
D. 1.06

Answer 1

Your problem doesn't make sense as stated...

Answer 2

@whpalmer4 If sin a = .25 and cos b = .25, which of the following is sin b/2 + cos a/2

Answer 3

I don't see anything following...

Answer 4

I just fixed the problem in the box. @whpalmer4

Answer 5

"which of the following is sin b/2 + cos a/2" — there's nothing following!

Answer 6

@whpalmer4
A. 1.60
B. 1.32
C. .26
D. 1.06

Answer 7

Do you know the half-angle identities for sin and cos?

Answer 8

\[\sin^2 u = \frac{1-\cos 2u}{2}\]\[\sin u = \sqrt{\frac{1-\cos 2u}{2}}\]
\[\cos^2 u = \frac{1+\cos 2u}{2}\]\[\cos u = \sqrt{\frac{1+\cos 2u}{2}}\]

Answer 9

Next, we draw the two triangles representing \(\sin a = 0.25\) and \(\cos b = 0.25\)



We'll use the triangles to find the value of \(\cos a\) and \(\sin b\) if needed.

Answer 10

We want \(\sin(b/2) + \cos(a/2)\) which we can construct with the identities given above if we let \(u = 0.25/2\). Then we'll have

\[\sin(b/2) + \cos(a/2) = \sqrt{\frac{1-\frac{1}{4}}{2}} + \sqrt{\frac{1+\cos(a)}{2}} = \sqrt{\frac{3}{8}} + \sqrt{\frac{1+\cos(a)}{2}}\]

Looking at that triangle we drew, we see that \(\cos(a) = \dfrac{\sqrt{15}}{4}\) so our equation becomes
\[\sin(b/2) + \cos(a/2) =\sqrt{\frac{3}{8}} + \sqrt{\frac{1+\frac{\sqrt{15}}{4}}{2}} = \sqrt{\frac{3}{8}}+ \sqrt{\frac{4+\sqrt{15}}{8}}\]
Punch that into your calculator and compare with your answer choices...