Question

Given sin θ = -3/5 and csc θ = -5/3 in quadrant III, find the value of other trigonometric functions using a Pythagorean Identity. Show your work.
Part I: Find the value of cos θ and sec θ

Part II: Using your answers from Part I, find the value of tan θ

Answer 1

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Answer 2

I. Since $\theta$ lies in quadrant 3, we know $\cos\theta$. The Pythagorean identity tells us

$\cos^2\theta+\sin^2\theta=1\implies\cos\theta=-\sqrt{1-\sin^2\theta}=\boxed{-\dfrac45}$

$\implies\sec\theta=\dfrac1{\cos\theta}=\boxed{-\dfrac54}$

II. By definition of tangent,

$\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac{-\frac35}{-\frac45}=\boxed{\dfrac34}$