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

Problem

cos(θ)=1/2,θ∈Quadrant I

Solution

1. Identify the given values for the right triangle using the definition cos(θ)=x/r

x=1

r=2

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

1+y2=2

1+y2=4

y2=3

y=√(,3)

Note: y is positive because θ is in Quadrant I.

3. Determine the sine value using the ratio sin(θ)=y/r

sin(θ)=√(,3)/2

4. Determine the tangent value using the ratio tan(θ)=y/x

tan(θ)=√(,3)/1=√(,3)

5. Determine the cosecant value by taking the reciprocal of sine.

csc(θ)=2/√(,3)

csc(θ)=(2√(,3))/3

6. Determine the secant value by taking the reciprocal of cosine.

sec(θ)=2/1=2

7. Determine the cotangent value by taking the reciprocal of tangent.

cot(θ)=1/√(,3)

cot(θ)=√(,3)/3

Final Answer

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

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