Question

If cos θ =-2/5 and tan θ > 0, what is the value of sin θ?

Answer 1

If the value of cos(θ) is negative, this angle (θ) can only be in one of two quadrants; the sine quadrant or the tan quadrant. In the sine quadrant, tan(θ) must be negative, but since tan(θ) > 0, we can safely say that the angle (θ) is based in the tan quadrant.

We know that cos(θ) = - Adjacent / Hypotenuse, and in this case Adjacent = 2 and Hypotenuse = 5. Using Pythagoras' theorem, we can find the opposite side of the right angled triangle situated in the tan quadrant...

Adjacent² + Opposite² = Hypotenuse²

Therefore:

2² + Opposite² = 5²

Opposite² = 5² - 2²

Opposite² = 21

Opposite = √(21)

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Now, sin(θ) must be negative, as the right angled triangle is in the tan quadrant. We also know that sin(θ) = Opposite / Hypotenuse, therefore:

sin(θ) = - [√(21)]/[5]

Answer 2

Value of sinθ = $\frac{-\sqrt{21} }{5}$ ,when cosθ = -2/5 and tanθ> 0 .

What is trigonometric function?

" Trigonometric function related to the sides of any triangle. There are six types of trigonometric function such as sinθ, cosθ, tanθ, cosec, sec and cot."

What is sinθ?

" Sinθ is equals to the ratio of opposite side and its hypotenuse."

Formula used

$(Adjacent side)^{2} + (opposite side)^{2} =(hypotenuse)^{2}$                ___(1)

cosθ = Adjacent side / Hypotenuse

sinθ = Opposite side / Hypotenuse

According to the question,

cosθ = $\frac{-2}{5}$  and tanθ >0

Adjacent side = -2                                      ___( 2)

Hypotenuse = 5                                         ____(3)

Substitute the value of (2) ,(3) in (1) we get,

$(-2)^{2} + (opposite side)^{2} =5^{2}$

⇒ (opposite side)² = 25 - 4

⇒ opposite side =±√21

As tanθ >0  ⇒ sinθ <0

Therefore,

sinθ = - √21 / 5

Hence the value sinθ = $\frac{-\sqrt{21} }{5}$ ,when cosθ = -2/5 and tanθ> 0 .

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