Question

Given cscθ = 3, find cos(θ).

Answer 1

`\csc\theta=(1)/(\sin\theta)\\\\\csc\theta=3\to(1)/(\sin\theta)=3\to\sin\theta=(1)/(3)\\\\\text{Use:}\ \sin^2x+\cos^2x=1\\\\\text{substitute}\\\\\left((1)/(3)\right)^2+\cos^2\theta=1\\\\(1)/(9)+\cos^2\theta=1\ \ \ \ |-(1)/(9)\\\\\cos^2\theta=(8)/(9)\to\cos\theta=\pm\sqrt{(8)/(9)}`

`\cos\theta=\pm(\sqrt8)/(\sqrt9)\to\cos\theta=\pm(√(4\cdot2))/(3)\to\cos\theta=\pm(\sqrt4\cdot\sqrt2)/(3)\\\\\boxed{\cos\theta=-(2√(2))/(3)\ \vee\ \cos\theta=(2\sqrt2)/(3)} }`

Answer 2

Remark

The first thing to do is to find the sin(theta). When that is done, use an identity to find the cosine.

Step One

Find Sin(theta)

Sin(theta) = 1/csc(theta)

Csc(theta) = 3

Sin(theta) = 1/3

Step Two

Find the first quadrant value of Cos(theta)

Cos(theta) = sqrt(1 - sin^2(theta) )

Cos(theta) = sqrt(1 - (1/3)^2 )

Cos(theta) = sqrt(1 - 1/9)

Cos(theta) = sqrt(8/9)

Cos(theta) = sqrt(8)/sqrt(9)

Cos(theta) = 2*sqrt(2) / 3 <<<<< Answer for quad One

Step Three

Find Cos(theta) for all 4 quads.

Quad 4

Quad 4 is the same as quad 1

Quads 2 and 3

Cos(theta) = a minus value. In this case cos(theta) = - 2sqrt(2) / 3 It's the same value, just minus.

Answer Cos(theta) = - 2 sqrt(2) / 3