Question

Find the exact value

Answer 1

Find the exact value of \[\cos (\theta + \phi)\] given that \[\cos \theta = \frac{ 12 }{ 13 } and \cos \phi= \frac{ 4 }{ 5 }\] and that \[\theta and \phi\] are between \[0 and \frac{ \pi }{ 2 }\]

Answer 2

you need the sine of both of those angles

Answer 3

then you can use
\[\cos (\theta + \phi)=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)\] so what is missing is
\(\sin(\theta)\) and \(\sin(\phi)\)
do you know how to find them?

Answer 4

these are sides of very familiar right triangles

Answer 5

draw a triangle and use Pythagorean therom?

Answer 6

exactly
but as i said, these are familiar right triangles with integer sides, so finding them might come from memory

Answer 7

And then I use the sum and difference formula correct?

Answer 8

yes, you can use the formula i wrote above

Answer 9

\[\cos (\theta + \phi)=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)\]
\[=\frac{12}{13}\times \frac{4}{5}-\frac{5}{13}\times\frac{3}{5}\] then some arithmetic

Answer 10

i jumped the gun and told you what the answers were to the other side of the triangles, but i assume you have seen a 3 - 4 - 5 right triangle and a 5 - 12 - 13 right triangle, so those numbers were easy enough to find

Answer 11

Okay, I got it!
Would I use the same method to solve:
Given that \[\sin \theta = -\frac{ 4 }{ 5 }\] with theta in quadrant IV,find \[\sin 2\]

Answer 12

yes, sorry i got locked out
that is right, what you said, use
\[\sin(2\theta)=2\sin(\theta)\cos(\theta)\]

Answer 13

Does the restriction of quadrant 4 have any affect on the answer?

Answer 14

And why did I use the sum and difference formula in the first equation and a double angle formula in this one?