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abruzzese [7]
3 years ago
14

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]

Mathematics
1 answer:
Lunna [17]3 years ago
8 0

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

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