Question

Please help me!!!

Answer 1

What have you tried so far?

Answer 2

I don't know what you are familiar with. Do you know the value of \(\cos(\pi/12)\)?

Answer 3

I typed I into a calculator and got .99, but I just wanted to see if it was the right answer

Answer 4

And yes I am familiar with cos(π/12)

Answer 5

I got .26 but the key in question is "exact value", so you may need to solve it by hand. Do you have any idea what to do?

You can try apply Angle-Sum Identity:

\(\cos(α + β) = \cos(α)\cos(β)-\sin(α)\sin(β) \)

You can let \(\alpha = \dfrac{4\pi}{12} = \dfrac{\pi}{3}\) and \(\beta = \dfrac{\pi}{12}\). Can you do it?

Answer 6

\(\alpha+\beta = \dfrac{5\pi}{12}\)

Answer 7

Is this kinda what you mean?

Answer 8

Oh yeah, that works too.

Answer 9

So, where do I go from there?

Answer 10

Use Angle-Sum Identity, with \(\alpha = \dfrac{\pi}{4}\) and \(\beta = \dfrac{\pi}{6}\)

Answer 11

hmm

Answer 12

\(\bf \cfrac{5}{12}\implies \cfrac{3+2}{12} \implies \cfrac{3}{12}+\cfrac{2}{12}\implies \cfrac{1}{4}+\cfrac{1}{6}\qquad thus
\\ \quad \\
cos\left( \frac{5\pi }{12} \right)\implies cos\left( \frac{3\pi +2\pi }{12} \right)\implies cos\left( \frac{1\pi }{4}+\frac{1\pi }{6} \right)
\\ \quad \\
cos({\color{brown}{ \alpha}} + {\color{blue}{ \beta}})= cos({\color{brown}{ \alpha}})cos({\color{blue}{ \beta}})- sin({\color{brown}{ \alpha}})sin({\color{blue}{ \beta}})\qquad thus
\\ \quad \\

cos\left( {\color{brown}{ \frac{\pi }{4}}}+{\color{blue}{ \frac{\pi }{6} }} \right)= cos({\color{brown}{ \frac{\pi }{4}}})cos({\color{blue}{ \frac{\pi }{6}}})- sin({\color{brown}{ \frac{\pi }{4}}})sin({\color{blue}{ \frac{\pi }{6}}})\)

Answer 13

plug and chug