### Home > PC > Chapter 4 > Lesson 4.1.2 > Problem4-29

4-29.

You can use the image of a BOW TIE to remind you to always draw a vertical line to the $x$-axis when completing a triangle to compute sine and cosine. This is because the triangle you work with will always be part of the bow tie.

In the diagram, assume the bow tie is symmetric with respect to both coordinate axes. Let $α = ∠$EOW, $β = ∠$EOB, $γ = ∠$EOT, and $δ = ∠$EOI, where all of these angles are in standard position and between $0$ and $2\pi$. The angles for $α$ and $β$ are shown. Assume sin $α =\frac { 1 } { 3 }$.

1. Without finding $α$, find the exact values for $\text{sin }β$, $\text{sin }γ$, and $\text{sin }δ$.

2. Use the Fundamental Pythagorean Identity or a right triangle to find $\text{cos }α$.

3. Find the exact values for $\text{cos }β$, $\text{cos }γ$, and $\text{cos }δ$.

$\frac{1}{3}$

$\frac{1}{3}$

$-\frac{1}{3}$

$-\frac{1}{3}$

$\text{sin}^2\theta+\text{cos}^2\theta=1$

$\left(\frac{1}{3}\right)^2+\text{cos}^2\theta=1$

$\text{cos}^2\theta=1-\left(\frac{1}{9}\right)$

$\text{cos}^2\theta=\frac{9}{9}-\left(\frac{1}{9}\right)$

$\text{cos}^2\theta=\frac{8}{9}$

$\text{cos}\theta=\sqrt\frac{8}{9}=\frac{\sqrt{8}}{3}=\frac{2\sqrt{2}}{3}$

The first two are negative $\text{cos}θ$.

The last one is positive $\text{cos}θ$.