Find the Other Trig Values in Quadrant II cos(theta)=-4/5

Problem

cos(θ)=−4/5,θ∈Quadrant II

Solution

1. Identify the given values for the reference triangle. Since cos(θ)=x/r we have x=−4 and r=5

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

(−4)2+y2=5

16+y2=25

y2=9

3. Determine the sign of y based on the quadrant. In Quadrant II, the ycoordinate is positive, so y=3

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

sin(θ)=3/5

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

tan(θ)=3/(−4)=−3/4

6. Calculate the reciprocal functions by inverting the primary ratios.

csc(θ)=1/sin(θ)=5/3

sec(θ)=1/cos(θ)=−5/4

cot(θ)=1/tan(θ)=−4/3

Final Answer

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

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