Question

Evaluate the expression by sketching a triangle. Sin(2cos^-1 3/5).

Answer 1

let's start with \(\large cos^{-1}(\frac{3}{5})=y \).
this means cosy = 3/5, correct? can you draw a triangle depicting this situation?

Answer 2

going to take more than just a triangle i think

Answer 3

we'll get to the double angle formula when we get there...

Answer 4

you need to know that \(\sin(2\theta)=2\sin(\theta )\cos(\theta)\)

Answer 5

cos=adj/hyp so the adj is 3 and the hypothenuse is 5

Answer 6

using pythagorean theorem the opposite side will be 4

Answer 7

but Idk what to do about the 2...

Answer 8

How can I solve this?

Answer 9

start with
\[\sin(2\theta)=2\sin(\theta )\cos(\theta)\] and so
\[\sin(\cos^{-1}(\frac{3}{5})=2\sin(\cos^{-1}(\frac{3}{5}))\times \cos(\cos^{-1}(\frac{3}{5}))\]

Answer 10

typo meant
\[\sin(2\cos^{-1}(\frac{3}{5})=2\sin(\cos^{-1}(\frac{3}{5}))\times \cos(\cos^{-1}(\frac{3}{5}))\]

Answer 11

one number is obvious
you know \(\cos(\cos^{-1}(\frac{3}{5}))=\frac{3}{5}\) that could be clear right?

Answer 12

yes

Answer 13

so what is left is to find \(\sin(\cos^{-1}(\frac{3}{5}))\) which is identical to saying
"if the cosine of the angle is \(\frac{3}{5}\) then what is the sine of the angle?

Answer 14

there is a picture of a triangle with the cosine of the angle \(\frac{3}{5}\)
what you need is the opposite side, which by pythagoras or the famous 3-4-5 right triangle must be 4

Answer 15

now we see that the sine of that angle is \(\frac{4}{5}\) so we have all the numbers we need

Answer 16

it is
\[\sin(2\cos^{-1}(\frac{3}{5}))=2\sin(\cos^{-1}(\frac{3}{5}))\times \cos(\cos^{-1}(\frac{3}{5}))\]
\[=2\times \frac{4}{5}\times \frac{3}{5}\] for your answer

Answer 17

hope steps are clear

Answer 18

yes, somewhat, thank you! =)