Find the Other Trig Values in Quadrant I sin(theta)=3/5

Problem

sin(θ)=3/5,θ∈Quadrant I

Solution

1. Identify the given information and the quadrant. In Quadrant I, all trigonometric functions (sin() cos() tan() csc() sec() cot() are positive.

2. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to find cos(θ)

(3/5)2+cos2(θ)=1

9/25+cos2(θ)=1

cos2(θ)=1−9/25

cos2(θ)=16/25

cos(θ)=√(,16/25)

cos(θ)=4/5

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

tan(θ)=(3/5)/(4/5)

tan(θ)=3/4

4. Find the reciprocal functions csc(θ) sec(θ) and cot(θ) by inverting the values of sin(θ) cos(θ) and tan(θ)

csc(θ)=1/sin(θ)=5/3

sec(θ)=1/cos(θ)=5/4

cot(θ)=1/tan(θ)=4/3

Final Answer

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

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