Find the Other Trig Values in Quadrant I sin(theta)=1/2

Problem

sin(θ)=1/2,θ∈Quadrant I

Solution

1. Identify the given trigonometric ratio and the quadrant. In Quadrant I, all trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are positive.

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

(1/2)2+cos2(θ)=1

1/4+cos2(θ)=1

cos2(θ)=3/4

cos(θ)=√(,3/4)

cos(θ)=√(,3)/2

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

tan(θ)=(1/2)/(√(,3)/2)

tan(θ)=1/√(,3)

tan(θ)=√(,3)/3

4. Determine the reciprocal identities for cosecant, secant, and cotangent.

csc(θ)=1/sin(θ)=2

sec(θ)=1/cos(θ)=2/√(,3)=(2√(,3))/3

cot(θ)=1/tan(θ)=√(,3)

Final Answer

cos(θ)=√(,3)/2,tan(θ)=√(,3)/3,csc(θ)=2,sec(θ)=(2√(,3))/3,cot(θ)=√(,3)

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