Find the Other Trig Values in Quadrant I sin(theta)=( square root of 5)/10

Problem

sin(θ)=√(,5)/10,θ∈Quadrant I

Solution

1. Identify the given values for the right triangle definition of sine, where sin(θ)=y/r

sin(θ)=√(,5)/10

y=√(,5)

r=10

2. Apply the Pythagorean theorem x2+y2=r2 to find the adjacent side x

x2+(√(,5))2=10

x2+5=100

x2=95

x=√(,95)

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

3. Determine the cosine value using cos(θ)=x/r

cos(θ)=√(,95)/10

4. Determine the tangent value using tan(θ)=y/x and rationalize the denominator.

tan(θ)=√(,5)/√(,95)

tan(θ)=√(,5)/(√(,5)⋅√(,19))

tan(θ)=1/√(,19)

tan(θ)=√(,19)/19

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

csc(θ)=10/√(,5)

csc(θ)=(10√(,5))/5

csc(θ)=2√(,5)

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

sec(θ)=10/√(,95)

sec(θ)=(10√(,95))/95

sec(θ)=(2√(,95))/19

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

cot(θ)=√(,19)

Final Answer

cos(θ)=√(,95)/10,tan(θ)=√(,19)/19,csc(θ)=2√(,5),sec(θ)=(2√(,95))/19,cot(θ)=√(,19)

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