Find the Other Trig Values in Quadrant III sin(theta)=-12/13

Problem

sin(θ)=−12/13,θ∈Quadrant III

Solution

1. Identify the given information and the signs of trigonometric functions in Quadrant III. In Quadrant III, both x and y coordinates are negative, meaning sin(θ)<0 and cos(θ)<0 while tan(θ)>0

2. Determine the values of y and r using the definition sin(θ)=y/r We have y=−12 and r=13

3. Calculate the value of x using the Pythagorean identity x2+y2=r2

x2+(−12)2=13

x2+144=169

x2=25

x=−5

(We choose x=−5 because x must be negative in Quadrant III).

4. Find the value of cos(θ) using the ratio x/r

cos(θ)=(−5)/13=−5/13

5. Find the value of tan(θ) using the ratio y/x

tan(θ)=(−12)/(−5)=12/5

6. Find the reciprocal identities for the remaining functions.

csc(θ)=1/sin(θ)=−13/12

sec(θ)=1/cos(θ)=−13/5

cot(θ)=1/tan(θ)=5/12

Final Answer

cos(θ)=−5/13,tan(θ)=12/5,csc(θ)=−13/12,sec(θ)=−13/5,cot(θ)=5/12

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