Question

QuestionDetermine the exact values of sin 0 and cos 0 given that the terminal side of angle 0 intersects the unit circle in the firstquadrant at (4/15,y).

Answer 1

Given a point on the circle at (4/15, y).

In a unit circle, the endpoint (x, y) is equal to (cos θ, sin θ).

Since x = cos θ and x = 4/15, then cos θ = 4/15.

Now, we know that cos θ = adjacent/hypotenuse, therefore, adjacent (x) = 4 and hypotenuse (r) = 15.

Let's solve for the opposite side (y) using the Pythagorean Theorem.

`\begin{gathered} r^2=x^2+y^2 \\ 15^2=4^2+y^2 \\ 225=16+y^2 \\ 225-16=y^2 \\ 209=y^2 \\ \sqrt[]{209}=\sqrt[]{y^2} \\ \sqrt[]{209}=y \end{gathered}`

The value of the opposite side is √209.

Now, we know that sin θ = opposite over hypotenuse therefore, sin θ =√209/15

`\sin \theta=\frac{\sqrt[]{209}}{15}`

To summarize, here are the values of cos θ and sin θ.

`\begin{gathered} \cos \theta=(4)/(15) \\ \sin \theta=\frac{\sqrt[]{209}}{15} \end{gathered}`