Chapter 7: Q. 15 (page 495)

Use the information given about the angle $0 < θ < 2 π$ , to find the exact value of
$a . \sin 2 θ b . \cos 2 θ c . \sin \frac{θ}{2} d . \cos \frac{θ}{2} e . \tan (2 θ) f . \tan \frac{θ}{2} c o t θ = - 2, s e c θ < 0$

Short Answer

Expert verified

Part (a). $\sin 2 θ = \frac{- 4}{5}$

part (b). $\cos 2 θ = \frac{3}{5}$

part (c). role="math" localid="1646651000338" $\sin \frac{θ}{2} = \sqrt{\frac{\sqrt{5} + 2}{2 \sqrt{5}}}$

part(d). $\cos \frac{θ}{2} = \sqrt{\frac{\sqrt{5} - 2}{2 \sqrt{5}}}$

part (e). $\tan 2 θ = \frac{- 4}{3}$

part (f). $\tan \frac{θ}{2} = \sqrt{\frac{\sqrt{5} + 2}{\sqrt{5} - 2}}$

Step by step solution

Part (a) Step 1. Value of sin2θ

We know that,

$\tan θ = \frac{1}{c o t θ} \tan θ = \frac{- 1}{2}$

$θ$ is in the second quadrant.

So,

$\sin θ = \frac{1}{\sqrt{5}} \cos θ = - \frac{2}{\sqrt{5}}$

Also,

$\sin 2 θ = 2 \sin θ \cos θ = 2 \times \frac{1}{\sqrt{5}} \times (- \frac{2}{\sqrt{5}}) = \frac{- 4}{5}$

part (b) Step 1. Value of cos2θ

We know that,

$\cos 2 θ = \cos^{2} θ - \sin^{2} θ = {(- \frac{2}{\sqrt{5}})}^{2} - {(\frac{1}{\sqrt{5}})}^{2} = \frac{4}{5} - \frac{1}{5} = \frac{3}{5}$

Part (c) Step 1. Value of sinθ2

We know that,

$\sin \frac{θ}{2} = \sqrt{\frac{1 - \cos θ}{2}} = \sqrt{\frac{1 - (- \frac{2}{\sqrt{5}})}{2}} = \sqrt{\frac{1 + \frac{2}{\sqrt{5}}}{2}} = \sqrt{\frac{\sqrt{5} + 2}{2 \sqrt{5}}}$

Part (d) Step 1. Value of cosθ2

We know that,

$\cos \frac{θ}{2} = \sqrt{\frac{1 + \cos θ}{2}} = \sqrt{\frac{1 + (- \frac{\sqrt{2}}{5})}{2}} = \sqrt{\frac{\sqrt{5} - 2}{2 \sqrt{5}}}$

Part (e) Part 1. Value of tan2θ

We have :

$\sin 2 θ = - \frac{4}{5} \cos 2 θ = \frac{3}{5}$

We know that,

$\tan 2 θ = \frac{\sin 2 θ}{\cos 2 θ} = \frac{\frac{- 4}{5}}{\frac{3}{5}} = - \frac{4}{3}$

Part (f) Step 1. Value of tanθ2

We have:

$\sin \frac{θ}{2} = \sqrt{\frac{\sqrt{5} + 2}{2 \sqrt{5}}} \cos \frac{θ}{2} = \sqrt{\frac{\sqrt{5} - 2}{2 \sqrt{5}}}$

We know that,

$\tan \frac{θ}{2} = \frac{\sin \frac{θ}{2}}{\cos \frac{θ}{2}} = \frac{\frac{\sqrt{\sqrt{5} + 2}}{2 \sqrt{5}}}{\frac{\sqrt{\sqrt{5} - 2}}{2 \sqrt{5}}} = \frac{\sqrt{5} + 2}{\sqrt{5} - 2}$

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Use the information given about the angle $0 < θ < 2 π$ , to find the exact value of
$a . \sin 2 θ b . \cos 2 θ c . \sin \frac{θ}{2} d . \cos \frac{θ}{2} e . \tan (2 θ) f . \tan \frac{θ}{2} c o t θ = - 2, s e c θ < 0$

Short Answer

Step by step solution

Part (a) Step 1. Value of sin2θ

part (b) Step 1. Value of cos2θ

Part (c) Step 1. Value of sinθ2

Part (d) Step 1. Value of cosθ2

Part (e) Part 1. Value of tan2θ

Part (f) Step 1. Value of tanθ2

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Use the information given about the angle 0<θ<2π, to find the exact value of a.sin2θb.cos2θc.sinθ2d.cosθ2e.tan2θf.tanθ2cotθ=-2,secθ<0

Short Answer

Step by step solution

Part (a) Step 1. Value of sin2θ

part (b) Step 1. Value of cos2θ

Part (c) Step 1. Value of sinθ2

Part (d) Step 1. Value of cosθ2

Part (e) Part 1. Value of tan2θ

Part (f) Step 1. Value of tanθ2

One App. One Place for Learning.

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Use the information given about the angle $0 < θ < 2 π$ , to find the exact value of
$a . \sin 2 θ b . \cos 2 θ c . \sin \frac{θ}{2} d . \cos \frac{θ}{2} e . \tan (2 θ) f . \tan \frac{θ}{2} c o t θ = - 2, s e c θ < 0$