MaryGarnCastaly
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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

Respuesta :

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

Step-by-step explanation:  Given that for an angle [tex]\theta[/tex],

[tex]\cos \theta=-\dfrac{2}{5},~~~\tan \theta>0.[/tex]

We are to find the value of [tex]\sin \theta.[/tex]

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 [tex]\theta[/tex] lies in the Quadrant III.

We will be using the following trigonometric identity:

[tex]\cos^2\theta+\sin^2\theta=1.[/tex]

We have

[tex]\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}.[/tex]

Since the angle [tex]\theta[/tex] lies in Quadrant III and sine is negative in that quadrant, so

[tex]\sin\theta=-\dfrac{\sqrt{21}}{5}.[/tex]

Thus, option (B) is correct.

Answer:

[tex]sin\theta=-\frac{\sqrt{21}}{5}[/tex] Option B is the correct option.

Step-by-step explanation:

It is given in the question [tex]cos\theta =-\frac{2}{5}[/tex] and [tex]tan\theta >0[/tex]

We have to calculate the value of [tex]sin\theta[/tex]

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 [tex]tan\theta >0[/tex] 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 [tex]sin\theta=-\frac{\sqrt{21}}{5}[/tex] will be the answer.

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