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If cos Θ = negative two over five and tan Θ > 0, what is the value of sin Θ? negative square root / 21 negative square root / 21 over 5 square root / 21 over 5 square root of 21 over 2

Verified Answer :

Answer:  The correct option is (B) negative square root 21 over 5.

Step-by-step explanation:  Given that for an angle \(\theta\),

\(\cos \theta=-\dfrac{2}{5},~~~\tan \theta>0.\)

We are to find the value of \(\sin \theta.\)

We know that

Cosine is negative in Quadrant II and Quadrant III.

Tangent is positive in Quadrant 1 and Quadrant III.

So, the given angle \(\theta\) lies in the Quadrant III.

We will be using the following trigonometric identity:

\(\cos^2\theta+\sin^2\theta=1.\)

We have

\(\cos \theta=-\dfrac{2}{5}\\\\\\\Rightarrow \sqrt{1-\sin^2\theta}=-\dfrac{2}{5}\\\\\\\Rightarrow 1-\sin^2\theta=\dfrac{4}{25}\\\\\\\Rightarrow \sin^2\theta=1-\dfrac{4}{25}\\\\\\\Rightarrow \sin^2\theta=\dfrac{21}{25}\\\\\\\Rightarrow \sin \theta=\pm\dfrac{\sqrt{21}}{5}.\)

Since the angle \(\theta\) lies in Quadrant III and sine is negative in that quadrant, so

\(\sin\theta=-\dfrac{\sqrt{21}}{5}.\)

Thus, option (B) is correct.

Answer:

\(sin\theta=-\frac{\sqrt{21}}{5}\) Option B is the correct option.

Step-by-step explanation:

It is given in the question \(cos\theta =-\frac{2}{5}\) and \(tan\theta >0\)

We have to calculate the value of \(sin\theta\)

We can see in the figure attached since cos∅ = -2/5

Here negative sign is showing that angle theta is lying either in quadrant 2nd or 3rd.

And \(tan\theta >0\) is showing as angle is either in quadrant 1 or 3rd.

Therefore it is confirm that angle is lying in quadrant 3 where sine is always negative.

Therefore \(sin\theta=-\frac{\sqrt{21}}{5}\) will be the answer.

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