Given: sin⁡θ=1517 and π2&lt;θ&lt;πFind the exact value of: cos⁡θ2A) 53434 New York CityB) −53434 Montreal, CanadaC) 33434 Barcelona, SpainD) −33434 Nassau, Bahamas

Step 1Ans: Given, sin⁡θ=1517π2&lt;θ&lt;πWe have to find exact value of cos⁡θ2Since, sin⁡θ=Perpendica―rHypoteνse⇒Perpendica―r=15Hypoteνse=17We know that, Pythagorean theoremHypoteνse2=Perpendica―r2+Base2⇒(17)2=(15)2+Base2⇒Base2=289−225=64⇒Base=8Since,cos⁡θ=BaseHypoteνse⇒cos⁡θ=817∵π2&lt;θ&lt;π(2ndrant)So,cos⁡θ=−817Now,We know thatcos2θ=2cos2θ−1⇒cos2θ=1+cos2θ2⇒cos2θ2=1+cos⁡θ2⇒cos⁡θ2=1+cos⁡θ2∵π2&lt;θ&lt;π⇒π4&lt;θ2&lt;π2(1strant)⇒cos⁡θ2=1+(−817){2}=17−817×2=934=334=334×3434=33434Hense, cos⁡θ2=334(34)⇒ Barselona, Spain

Check the solution to "Given: sin⁡θ=1517 and π2<θ<πFind the exact value of:..."

High School
Geometry
Trigonometry
Right triangles and trigonometry

divamannyxv

Open question

2022-08-17

Given:

\sin θ = \frac{15}{17}

and

\frac{π}{2} < θ < π

Find the exact value of:

\cos \frac{θ}{2}

\frac{5 \sqrt{34}}{34}

New York City
B)

- \frac{5 \sqrt{34}}{34}

Montreal, Canada
C)

\frac{3 \sqrt{34}}{34}

Barcelona, Spain
D)

- \frac{3 \sqrt{34}}{34}

Nassau, Bahamas

Answer & Explanation

Kristen Garrison

Beginner2022-08-18Added 11 answers

Step 1
Ans: Given,

\sin θ = \frac{15}{17}

\frac{π}{2} < θ < π

We have to find exact value of

\cos \frac{θ}{2}

Since,

\sin θ = \frac{P e r p e n d i c \underset{―}{a} r}{H y p o t e ν s e}

\Rightarrow P e r p e n d i c \underset{―}{a} r = 15

H y p o t e ν s e = 17

We know that,
Pythagorean theorem

H y p o t e ν s e^{2} = P e r p e n d i c \underset{―}{a} r^{2} + B a s e^{2}

\Rightarrow {(17)}^{2} = {(15)}^{2} + B a s e^{2}

\Rightarrow B a s e^{2} = 289 - 225 = 64

\Rightarrow B a s e = 8

Since,

\cos θ = \frac{B a s e}{H y p o t e ν s e}

\Rightarrow \cos θ = \frac{8}{17}

∵ \frac{π}{2} < θ < π (2^{n} d r a n t)

So,

\cos θ = \frac{- 8}{17}

Now,
We know that

\cos^{2} θ = 2 \cos^{2} θ - 1

\Rightarrow \cos^{2} θ = \frac{1 + \cos^{2} θ}{2}

\Rightarrow \cos^{2} \frac{θ}{2} = \frac{1 + \cos θ}{2}

\Rightarrow \cos \frac{θ}{2} = \sqrt{\frac{1 + \cos θ}{2}}

∵ \frac{π}{2} < θ < π \Rightarrow \frac{π}{4} < \frac{θ}{2} < \frac{π}{2} (1^{s t} r a n t)

\Rightarrow \cos \frac{θ}{2} = \sqrt{\frac{1 + (\frac{- 8}{17}) {2}}{=} \sqrt{\frac{17 - 8}{17 \times 2}} = \sqrt{\frac{9}{34}} = \frac{3}{\sqrt{34}} = \frac{3}{\sqrt{34}} \times \frac{\sqrt{34}}{\sqrt{34}} = \frac{3 \sqrt{34}}{34}}

Hense,

\cos \frac{θ}{2} = \frac{3 \sqrt{34}}{(34)}

\Rightarrow

Barselona, Spain

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