Find the Other Trig Values in Quadrant III cos(theta)=-5/13

Problem

cos(θ)=−5/13,θ∈Quadrant III

Solution

1. Identify the given values for the adjacent side x and the hypotenuse r using the definition cos(θ)=x/r

x=−5

r=13

2. Apply the Pythagorean theorem x2+y2=r2 to find the vertical component y

(−5)2+y2=13

25+y2=169

y2=144

3. Determine the sign of y based on the quadrant. Since θ is in Quadrant III, both x and y must be negative.

y=−12

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

sin(θ)=−12/13

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

tan(θ)=(−12)/(−5)=12/5

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

csc(θ)=1/sin(θ)=−13/12

sec(θ)=1/cos(θ)=−13/5

cot(θ)=1/tan(θ)=5/12

Final Answer

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

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