If cos θ =-2/5 and tan θ > 0, what is the value of sin θ?
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]
Value of sinθ = \(\frac{-\sqrt{21} }{5}\) ,when cosθ = -2/5 and tanθ> 0 .
" 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."
" 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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