Q60 / 160 · AP EAMCET 2023 (17 May Shift 2) · Mathematics

If $\sin \theta=\frac{3}{5}$ and $\theta$ is not in the first quadrant, then $15 \sin$ $2 \theta-20 \cos 2 \theta-7 \tan 2 \theta=$

  • A
    -4
  • B
    -12
  • C
    12
  • Correct answer
    D
    4

Solution

Given: $\sin \theta=\frac{3}{5}$ Since, $\theta$ is not in first quadrant then $\theta$ will be in second quadrant because $\sin \theta$ is positive in first \& second coordinate. $\cos \theta=\sqrt{1-\sin ^2 \theta}=-\sqrt{1-\frac{9}{25}}=-\frac{4}{5}$ Now, $\sin 2 \theta=2 \sin \theta \cdot \cos \theta=2 \cdot \frac{3}{5}\left(-\frac{9}{5}\right)=-\frac{24}{25}$ $\begin{aligned} & \cos 2 \theta=2 \cos ^2 \theta-1=2\left(-\frac{4}{5}\right)^2-1=\frac{7}{25} \\ & \tan 2 \theta=-\frac{24}{7}\end{aligned}$ Now, $15 \sin 2 \theta-20 \cos 2 \theta-7 \tan 2 \theta$ $=15 .\left(-\frac{24}{25}\right)-20 \times \frac{7}{25}-7\left(-\frac{24}{7}\right)=4$

Hence, the correct answer is option: 'D'.

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