Find the Other Trig Values in Quadrant III cos(theta)=-24/25

Problem

cos(θ)=−24/25,θ∈Quadrant III

Solution

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

x=−24

r=25

2. Apply the Pythagorean theorem x^2+y^2=r^2 to find the vertical component y

(−24)^2+y^2=25^2

576+y^2=625

y^2=49

3. Determine the sign of y based on the location of θ in Quadrant III, where both x and y are negative.

y=−7

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

sin(θ)=−7/25

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

tan(θ)=(−7)/(−24)=7/24

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

csc(θ)=1/sin(θ)=−25/7

sec(θ)=1/cos(θ)=−25/24

cot(θ)=1/tan(θ)=24/7

Final Answer

sin(θ)=−7/25,tan(θ)=7/24,csc(θ)=−25/7,sec(θ)=−25/24,cot(θ)=24/7

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