If cos θ = -5/13 and θ is in the third quadrant, what is the value of tan θ?
Show the answer and explanation
Correct answer
A. 12/5
Principle or equation
In the third quadrant, both sine and cosine are negative, and tangent is positive. Use the Pythagorean identity sin²θ + cos²θ = 1 to find sin θ, then tan θ = sin θ / cos θ.
Why this answer is correct
cos θ = -5/13. Then sin²θ = 1 - (25/169) = 144/169, so sin θ = ±12/13. Since θ is in the third quadrant, sin θ = -12/13. Therefore tan θ = (-12/13) / (-5/13) = 12/5.
Example
If cos θ = -3/5 in quadrant II, then sin θ = 4/5 and tan θ = -4/3.
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