# Given: theta in [pi /2, pi ] and tan theta = 1/3. Find sin theta and cos theta (but to find theangle is not needed)

Samson Kaufman 2022-08-04 Answered
Given: $\theta \in \left[\pi /2,\pi \right]$ and $\mathrm{tan}\theta =1/3$.
Find $\mathrm{sin}\theta$ and $\mathrm{cos}\theta$ (but to find the angle is not needed)
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pelvogp
Given: $\theta \in \left[\pi /2,\pi \right]$ and $\mathrm{tan}\theta =1/3$.
Find $\mathrm{sin}\theta$ and $\mathrm{cos}\theta$ (but to find theangle is not needed)
$\mathrm{tan}\theta =1/3⇒hyp=\sqrt{9+1}=\sqrt{10}$
$\mathrm{sin}\theta =1/\sqrt{10}$
$\mathrm{cos}\theta =3/\sqrt{10}$

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we have two solutions to the question
$\mathrm{tan}\theta =1/3$
clearly we have a right angle triangle with
p=1, b=3
thus $h=\sqrt{{3}^{2}}+{1}^{2}=±\sqrt{10}$
so either $\mathrm{cos}\theta =3/\sqrt{10},\mathrm{sin}\theta =1/\sqrt{10}$
or $\mathrm{cos}\theta =-3/\sqrt{10},\mathrm{sin}\theta =-1/\sqrt{10}$
if n is specified then $\theta$ and thus exact solution can be determined

We have step-by-step solutions for your answer!