Problem

Given that cos⁡(θ)=−35 and θ is in the second quadrant, find the values of sin⁡(θ), tan⁡(θ), csc⁡(θ), sec⁡(θ), and cot⁡(θ).

Answer

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Answer

Step 5: The reciprocal of tan⁡(θ) is cot⁡(θ), so cot⁡(θ)=1tan⁡(θ)=1−43=−34

Steps

Step 1 ：Step 1: We know that cos⁡(θ) is negative in the second quadrant, and the value of sin⁡(θ) is positive. We can use the Pythagorean identity sin2⁡(θ)+cos2⁡(θ)=1 to find sin⁡(θ). So sin2⁡(θ)=1−cos2⁡(θ)=1−(−35)2=1625, thus sin⁡(θ)=1625=45

Step 2 ：Step 2: We can find tan⁡(θ) using the identity tan⁡(θ)=sin⁡(θ)cos⁡(θ). So tan⁡(θ)=45−35=−43

Step 3 ：Step 3: The reciprocal of sin⁡(θ) is csc⁡(θ), so csc⁡(θ)=1sin⁡(θ)=145=54

Step 4 ：Step 4: The reciprocal of cos⁡(θ) is sec⁡(θ), so sec⁡(θ)=1cos⁡(θ)=1−35=−53

Step 5 ：Step 5: The reciprocal of tan⁡(θ) is cot⁡(θ), so cot⁡(θ)=1tan⁡(θ)=1−43=−34

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