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If tan(theta) = -1/2 and pi/2 < theta < pi, what is cos(theta)?

User WordSmith
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1 Answer

2 votes

Answer:


\cos \theta=-(2√(5))/(5)

Explanation:

It was given that:


\tan(\theta)=-(1)/(2)

and
(\pi)/(2)\le \theta \le \pi.

We use the relation:


\sec^2 \theta=1+\tan^2\theta

We substitute the value to get:


\sec^2 \theta=1+((1)/(2))^2


\sec^2 \theta=1+(1)/(4)


\sec^2 \theta=(5)/(4)


\sec \theta=\pm (√(5))/(2)

Reciprocate both sides to get


\cos \theta=\pm (2)/(√(5))

Rationalize the denominator:


\cos \theta=\pm (2√(5))/(5)

The given interval,
(\pi)/(2)\le \theta \le \pi is the same as the 2nd quadrant where the cosine ratio is negative.


\therefore \cos \theta=-(2√(5))/(5)

User AlexZvl
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