If tan θ = 3/4 and θ is in the third quadrant, what is the value of sin θ?
Show the answer and explanation
Correct answer
A. -3/5
Principle or equation
In the third quadrant, both sine and cosine are negative. Given tan θ = sin θ / cos θ, we can set sin θ = -3k and cos θ = -4k for some positive k. Using sin^2 θ + cos^2 θ = 1, find k.
Why this answer is correct
Let sin θ = -3k and cos θ = -4k. Then (-3k)^2 + (-4k)^2 = 9k^2 + 16k^2 = 25k^2 = 1, so k^2 = 1/25, k = 1/5. Thus sin θ = -3/5.
Example
If tan θ = 5/12 and θ is in the third quadrant, then sin θ = -5/13.
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