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Find the following trigonometric values. Express your answers exactly. {:[cos(240^(@))=],[sin(240^(@))=]:}

Find the following trigonometric values.\newlineExpress your answers exactly.\newlinecos⁡(240∘)=sin⁡(240∘)= \begin{array}{l} \cos \left(240^{\circ}\right)= \\ \sin \left(240^{\circ}\right)= \end{array}

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Q. Find the following trigonometric values.\newlineExpress your answers exactly.\newlinecos⁡(240∘)=sin⁡(240∘)= \begin{array}{l} \cos \left(240^{\circ}\right)= \\ \sin \left(240^{\circ}\right)= \end{array}
  1. Use Unit Circle: To find the exact values of cos⁡(240∘)\cos(240^\circ) and sin⁡(240∘)\sin(240^\circ), we need to use the unit circle and the properties of trigonometric functions in different quadrants. The angle of 240∘240^\circ is in the third quadrant, where cosine is negative and sine is also negative.
  2. Find cos⁡(240°)\cos(240°): First, let's find cos⁡(240°)\cos(240°). We know that cos⁡(240°)\cos(240°) is the same as cos⁡(240°−180°)\cos(240° - 180°), which is cos⁡(60°)\cos(60°), but since 240°240° is in the third quadrant, the cosine value will be negative.
  3. Find sin⁡(240°)\sin(240°): The exact value of cos⁡(60°)\cos(60°) is 12\frac{1}{2}. Therefore, cos⁡(240°)\cos(240°) is −12-\frac{1}{2}.
  4. Final Results: Now, let's find sin⁡(240°)\sin(240°). We know that sin⁡(240°)\sin(240°) is the same as sin⁡(240°−180°)\sin(240° - 180°), which is sin⁡(60°)\sin(60°), but since 240°240° is in the third quadrant, the sine value will be negative.
  5. Final Results: Now, let's find sin⁡(240°)\sin(240°). We know that sin⁡(240°)\sin(240°) is the same as sin⁡(240°−180°)\sin(240° - 180°), which is sin⁡(60°)\sin(60°), but since 240°240° is in the third quadrant, the sine value will be negative.The exact value of sin⁡(60°)\sin(60°) is 3/2\sqrt{3}/2. Therefore, sin⁡(240°)\sin(240°) is −3/2-\sqrt{3}/2.

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