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

Problem

cos(θ)=−3/5,θ∈Quadrant III

Solution

1. Identify the given values for the reference triangle. Since cos(θ)=x/r we have x=−3 and r=5

2. Apply the Pythagorean theorem to find the missing y coordinate using x2+y2=r2

(−3)2+y2=5

9+y2=25

y2=16

3. Determine the sign of y In Quadrant III, both x and y are negative, so y=−4

4. Calculate the sine value using sin(θ)=y/r

sin(θ)=−4/5

5. Calculate the tangent value using tan(θ)=y/x

tan(θ)=(−4)/(−3)=4/3

6. Calculate the reciprocal identities for cosecant, secant, and cotangent.

csc(θ)=1/sin(θ)=−5/4

sec(θ)=1/cos(θ)=−5/3

cot(θ)=1/tan(θ)=3/4

Final Answer

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

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