Question

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

Answer 1

`sin\theta=-(√(21))/(5)` Option B is the correct option.

Explanation:

It is given in the question
`cos\theta =-(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=-(√(21))/(5)` will be the answer.

Answer 2

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

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

`\cos \theta=-(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=-(2)/(5)\\\\\\\Rightarrow √(1-\sin^2\theta)=-(2)/(5)\\\\\\\Rightarrow 1-\sin^2\theta=(4)/(25)\\\\\\\Rightarrow \sin^2\theta=1-(4)/(25)\\\\\\\Rightarrow \sin^2\theta=(21)/(25)\\\\\\\Rightarrow \sin \theta=\pm(√(21))/(5).`

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

`\sin\theta=-(√(21))/(5).`

Thus, option (B) is correct.