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<0`. The Pythagorean identity tells us

`\cos^2\theta+\sin^2\theta=1\implies\cos\theta=-√(1-\sin^2\theta)=\boxed{-\frac45}`

`\implies\sec\theta=\frac1{\cos\theta}=\boxed{-\frac54}`

II. By definition of tangent,

`\tan\theta=(\sin\theta)/(\cos\theta)=(-\frac35)/(-\frac45)=\boxed{\frac34}`