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

Problem

sin(θ)=4/5,θ∈Quadrant II

Solution

1. Identify the given information and the quadrant. In Quadrant II, the ycoordinate (sine) is positive, while the xcoordinate (cosine) is negative.

2. Apply the Pythagorean identity sin^2(θ)+cos^2(θ)=1 to find the value of cos(θ)

(4/5)^2+cos^2(θ)=1

3. Solve for cos^2(θ) by squaring the fraction and subtracting it from 1.

16/25+cos^2(θ)=1

cos^2(θ)=1−16/25

cos^2(θ)=9/25

4. Determine the sign of cos(θ) Since θ is in Quadrant II, cos(θ) must be negative.

cos(θ)=−√(,9/25)

cos(θ)=−3/5

5. Calculate tan(θ) using the ratio sin(θ)/cos(θ)

tan(θ)=(4/5)/(−3/5)

tan(θ)=−4/3

6. Find the reciprocal identities for the remaining trigonometric functions: csc(θ)=1/sin(θ) sec(θ)=1/cos(θ) and cot(θ)=1/tan(θ)

csc(θ)=5/4

sec(θ)=−5/3

cot(θ)=−3/4

Final Answer

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

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