Question

Find the exact values of sin(2θ), cos(2θ), and tan(2θ) given sec(θ) = − 15/9 and θ is in the interval [pi/2, pi]

Answer 1

Right away,

$\sec\theta=-\dfrac{15}9\implies\cos\theta=-\dfrac9{15}=-\dfrac35$

Recall that $\sin^2\theta+\cos^2\theta=1$, so $\sin\theta=\pm\sqrt{1-\cos^2\theta}$. We're told that $\dfrac\pi2\le\theta\le\pi$, for which we expect to have $\sin\theta>0$ and so we take the positive square root. Also recall that 3-4-5 is a Pythagorean triple. Then

$\sin\theta=\dfrac45$

We have everything we need to find the remaining trig ratios. With the help of double angle identities, we get

$\sin2\theta=2\sin\theta\cos\theta=2\left(\dfrac45\right)\left(-\dfrac35\right)=-\dfrac{24}{25}$

$\cos2\theta=\cos^2\theta-\sin^2\theta=\left(-\dfrac35\right)^2-\left(\dfrac45\right)^2=-\dfrac7{25}$

$\tan2\theta=\dfrac{\sin2\theta}{\cos2\theta}=\dfrac{24}7$