Question

Find the exact value of tan(x/2) and sin 2x, given cos x=24/25 and sin x>0

Answer 1

have you covered the half-angle and double-angle trig identities yet?

Answer 2

Yes

Answer 3

@jdoe0001 Why?

Answer 4

ohh so you can use the half-angle identities

Answer 5

and double-angle ones

Answer 6

\(\bf cos(x)=\cfrac{24}{25}\implies \cfrac{adjacent}{hypotenuse}\implies \cfrac{a=24}{c=25}
\\ \quad \\
c^2=a^2+b^2\implies \pm\sqrt{c^2-a^2}={\color{blue}{ b}}\quad \textit{so which is it }\pm ?
\\ \quad \\
well\quad sin(x)>0\quad \textit{meaning sine is positive, so "b" is positive}\)

Answer 7

once you get the opposite side, or "b", then you can just use that in your trig double and half-angle identities
\(\bf tan\left(\frac{x}{2}\right)=\cfrac{sin(x)}{1+cos(x)}\implies tan\left(\frac{x}{2}\right)=\cfrac{\frac{{\color{blue}{ b}}}{c}}{1+\frac{a}{c}}
\\ \quad \\
sin(2x)=2sin(x)cos(x)\implies sin(2x)=2\cfrac{{\color{blue}{ b}}}{c}\cdot \cfrac{a}{c}\)

Answer 8

I don't understand how the tan(x/2) becomes sin(x)/1+cos(x) @jdoe0001

Answer 9

You may use trig identity: cos x = (1 - t^2)/(1 + t^2), with t = tan (x/2)
24 (1 + t^2) = 25 (1 - t^2)

Answer 10

So then what would my answer be? I'm confused.

Answer 11

hope its clear???

Answer 12

This is the one part that I truly have a lot of problems on, but you've made it a lot clearer! Thanks for all your help guys!!

Answer 13

welcome