Questions: Build a reference triangle below for θ=330° and use your labeled triangle to find sin θ and tan θ.

Transcript text: Build a reference triangle below for $\theta=330^{\circ}$ and use your labeled triang to find $\sin \theta$ and $\tan \theta$.

Solution

Solution Steps

Step 1: Determine the Reference Angle

The given angle is $ \theta = 330^\circ $. To find the reference angle, subtract $ 330^\circ $ from $ 360^\circ $: \[ 360^\circ - 330^\circ = 30^\circ \] So, the reference angle is $ 30^\circ $.

Step 2: Draw the Reference Triangle

Draw a right triangle in the fourth quadrant with the reference angle $ 30^\circ $. Label the sides of the triangle based on the standard 30-60-90 triangle ratios:

Opposite side to $ 30^\circ $: $ 1 $
Adjacent side to $ 30^\circ $: $ \sqrt{3} $
Hypotenuse: $ 2 $

Step 3: Determine the Signs of the Trigonometric Functions

Since $ 330^\circ $ is in the fourth quadrant:

Sine is negative.
Cosine is positive.
Tangent is negative.

Step 4: Calculate $ \sin \theta $ and $ \tan \theta $

Using the reference triangle: \[ \sin 330^\circ = -\sin 30^\circ = -\frac{1}{2} \] \[ \tan 330^\circ = -\tan 30^\circ = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \]