Find the Other Trig Values in Quadrant III sin(theta)=-3/5

Problem

sin(θ)=−3/5,θ∈Quadrant III

Solution

1. Identify the given information and the quadrant. In Quadrant III, both the xcoordinate (cosine) and ycoordinate (sine) are negative, while their ratio (tangent) is positive.

2. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find the value of cos(θ)

(−3/5)2+cos2(θ)=1

9/25+cos2(θ)=1

cos2(θ)=1−9/25

cos2(θ)=16/25

3. Solve for cosine by taking the square root. Since θ is in Quadrant III, cos(θ) must be negative.

cos(θ)=−√(,16/25)

cos(θ)=−4/5

4. Calculate tangent using the identity tan(θ)=sin(θ)/cos(θ)

tan(θ)=(−3/5)/(−4/5)

tan(θ)=3/4

5. Find the reciprocal identities for cosecant, secant, and cotangent.

csc(θ)=1/sin(θ)=−5/3

sec(θ)=1/cos(θ)=−5/4

cot(θ)=1/tan(θ)=4/3

Final Answer

cos(θ)=−4/5,tan(θ)=3/4,csc(θ)=−5/3,sec(θ)=−5/4,cot(θ)=4/3

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