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Find the following trigonometric values. Express your answers exactly. {:[cos(135^(@))=],[sin(135^(@))=]:}

Find the following trigonometric values.\newlineExpress your answers exactly.\newlinecos⁡(135∘)=sin⁡(135∘)= \begin{array}{l} \cos \left(135^{\circ}\right)= \\ \sin \left(135^{\circ}\right)= \end{array}

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Q. Find the following trigonometric values.\newlineExpress your answers exactly.\newlinecos⁡(135∘)=sin⁡(135∘)= \begin{array}{l} \cos \left(135^{\circ}\right)= \\ \sin \left(135^{\circ}\right)= \end{array}
  1. Using the Unit Circle: To find the exact values of cos⁡(135∘)\cos(135^\circ) and sin⁡(135∘)\sin(135^\circ), we can use the unit circle and the properties of trigonometric functions in different quadrants. The angle 135∘135^\circ is located in the second quadrant, where cosine is negative and sine is positive. We can also use the reference angle of 45∘45^\circ, which is the acute angle that 135∘135^\circ makes with the x-axis.
  2. Using the Reference Angle: The cosine and sine of 45∘45^\circ are known exact values. Since 135∘135^\circ is 45∘45^\circ more than 90∘90^\circ, we can use the fact that cos⁡(135∘)=−cos⁡(45∘)\cos(135^\circ) = -\cos(45^\circ) and sin⁡(135∘)=sin⁡(45∘)\sin(135^\circ) = \sin(45^\circ) because cosine is negative in the second quadrant and sine is positive.
  3. Finding cos⁡(135∘)\cos(135^\circ): The exact value of cos⁡(45∘)\cos(45^\circ) is 2/2\sqrt{2}/2. Therefore, cos⁡(135∘)=−cos⁡(45∘)=−2/2\cos(135^\circ) = -\cos(45^\circ) = -\sqrt{2}/2.
  4. Finding sin⁡(135°)\sin(135°): Similarly, the exact value of sin⁡(45°)\sin(45°) is also 2/2\sqrt{2}/2. Therefore, sin⁡(135°)=sin⁡(45°)=2/2\sin(135°) = \sin(45°) = \sqrt{2}/2.

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