Chapter 12: Q. 12 (page 840)

If $\sin θ = \frac{1}{4}$ and $θ$ is in second quadrant, find
(a) $\cos θ$ (b) $\tan θ$
(c) $\sin (2 θ)$ (d) $\cos (2 θ)$
(e) $\sin (\frac{1}{2} θ)$

Short Answer

Expert verified

(a) $\cos θ = - \frac{\sqrt{15}}{4}$

(b) $\tan θ = - \frac{1}{\sqrt{15}}$

(d) $\cos (2 θ) = \frac{7}{8}$

(e) $\sin (\frac{1}{2} θ) = \frac{\sqrt{4 + \sqrt{15}}}{2 \sqrt{2}}$

Step by step solution

Step 1. Given Information

Given $\sin θ = \frac{1}{4}$ and $\frac{π}{2} < θ < π$ .

Part (a) Step 1. Finding cosθ

We know that

$\cos^{2} θ + \sin^{2} θ = 1 \cos^{2} θ + {(\frac{1}{4})}^{2} = 1 \cos^{2} θ = 1 - \frac{1}{16} = \frac{15}{16} \cos θ = \pm \frac{\sqrt{15}}{4} \cos θ = - \frac{\sqrt{15}}{4}$

Part (b) Step 1. Finding tanθ

$\tan θ = \frac{1 / 4}{- \sqrt{15} / 4} \tan θ = \frac{1}{4} \times \frac{- 4}{\sqrt{15}} \tan θ = - \frac{1}{\sqrt{15}}$

Part (c) Step 1. Finding sin2θ

$\sin (2 θ) = 2 \times \frac{1}{4} \times - \frac{\sqrt{15}}{4} \sin (2 θ) = - \frac{\sqrt{15}}{8}$

Part (d) Step 1. Finding cos2θ

$\cos 2 θ = {(- \frac{\sqrt{15}}{4})}^{2} - {(\frac{1}{4})}^{2} \cos 2 θ = \frac{15}{16} - \frac{1}{16} = \frac{14}{16} = \frac{7}{8}$

Part (e) Step 1. Finding sin12θ

$\sin (\frac{1}{2} θ) = \sqrt{\frac{1 - (- \frac{\sqrt{15}}{4})}{2}} \sin (\frac{1}{2} θ) = \sqrt{\frac{1 + \frac{\sqrt{15}}{4}}{2}} \sin (\frac{1}{2} θ) = \sqrt{\frac{4 + \sqrt{15}}{8}} \sin (\frac{1}{2} θ) = \frac{\sqrt{4 + \sqrt{15}}}{2 \sqrt{2}}$

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If $\sin θ = \frac{1}{4}$ and $θ$ is in second quadrant, find
(a) $\cos θ$ (b) $\tan θ$
(c) $\sin (2 θ)$ (d) $\cos (2 θ)$
(e) $\sin (\frac{1}{2} θ)$

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Finding cosθ

Part (b) Step 1. Finding tanθ

Part (c) Step 1. Finding sin2θ

Part (d) Step 1. Finding cos2θ

Part (e) Step 1. Finding sin12θ

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If sinθ=14and θis in second quadrant, find(a) cosθ (b) tanθ(c) sin2θ (d) cos2θ(e)sin12θ

Short Answer

Step by step solution

Step 1. Given Information

Part (a) Step 1. Finding cosθ

Part (b) Step 1. Finding tanθ

Part (c) Step 1. Finding sin2θ

Part (d) Step 1. Finding cos2θ

Part (e) Step 1. Finding sin12θ

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

If $\sin θ = \frac{1}{4}$ and $θ$ is in second quadrant, find
(a) $\cos θ$ (b) $\tan θ$
(c) $\sin (2 θ)$ (d) $\cos (2 θ)$
(e) $\sin (\frac{1}{2} θ)$