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]