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Find the following trigonometric values. Express your answers exactly. {:[cos(330^(@))=],[sin(330^(@))=]:}

Find the following trigonometric values.\newlineExpress your answers exactly.\newlinecos⁡(330∘)=sin⁡(330∘)= \begin{array}{l} \cos \left(330^{\circ}\right)= \\ \sin \left(330^{\circ}\right)= \end{array}

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Q. Find the following trigonometric values.\newlineExpress your answers exactly.\newlinecos⁡(330∘)=sin⁡(330∘)= \begin{array}{l} \cos \left(330^{\circ}\right)= \\ \sin \left(330^{\circ}\right)= \end{array}
  1. Position on Unit Circle: Understand the position of 330∘330^\circ on the unit circle.\newline330∘330^\circ is in the fourth quadrant of the unit circle, where cosine values are positive and sine values are negative.
  2. Determine Reference Angle: Determine the reference angle for 330°330°.\newlineThe reference angle is the acute angle that the terminal side of the angle makes with the x-axis. For 330°330°, the reference angle is 360°−330°=30°360° - 330° = 30°.
  3. Find Exact Values: Use the reference angle to find the exact values of cosine and sine.\newlineSince the cosine and sine of 30∘30^\circ are known exact values, we can use them to find the values for 330∘330^\circ, keeping in mind the signs for the fourth quadrant.
  4. Calculate Cosine: Calculate the cosine of 330°330°. \newlinecos⁡(330°)=cos⁡(360°−30°)=cos⁡(30°)\cos(330°) = \cos(360° - 30°) = \cos(30°) because cosine is positive in the fourth quadrant.\newlineWe know that cos⁡(30°)=3/2\cos(30°) = \sqrt{3}/2.\newlineTherefore, cos⁡(330°)=3/2\cos(330°) = \sqrt{3}/2.
  5. Calculate Sine: Calculate the sine of 330°330°. \newlinesin⁡(330°)=sin⁡(360°−30°)=−sin⁡(30°)\sin(330°) = \sin(360° - 30°) = -\sin(30°) because sine is negative in the fourth quadrant.\newlineWe know that sin⁡(30°)=12\sin(30°) = \frac{1}{2}.\newlineTherefore, sin⁡(330°)=−12\sin(330°) = -\frac{1}{2}.

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