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Chapter 27: Problem 869

Find the sine and cosine of \(75^{\circ}\).

Short Answer

Expert verified
The sine of 75 degrees is \(\frac{\sqrt{3}+1}{2\sqrt{2}}\), and the cosine of 75 degrees is \(\frac{\sqrt{3}-1}{2\sqrt{2}}\).

Step by step solution

01

Split the angle

Split the given angle of 75 degrees into the sum of two known angles, 45 degrees, and 30 degrees: \( 75^{\circ} = 45^{\circ} + 30^{\circ} \)
02

Use sum of angles formulas

Use the sum of angles formulas for sine and cosine to find the sine and cosine of 75 degrees: Sine formula: \( \sin(A+B)=\sin A \cos B + \cos A \sin B \) Cosine formula: \( \cos(A+B)= \cos A \cos B - \sin A \sin B \)
03

Prepare known values

Use the known sine and cosine values for 45 and 30 degrees: \( \sin 45^{\circ} = \frac{1}{\sqrt{2}}\) \( \cos 45^{\circ} = \frac{1}{\sqrt{2}}\) \( \sin 30^{\circ} = \frac{1}{2}\) \( \cos 30^{\circ} = \frac{\sqrt{3}}{2}\)
04

Calculate sine and cosine of 75 degrees

Use the formulas from Step 2 and the values from Step 3 to find the sine and cosine of 75 degrees: \(\sin 75^{\circ}=\sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ}\) \(\sin 75^{\circ}=\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2}\) \(\cos 75^{\circ}= \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ}\) \(\cos 75^{\circ}= \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\)
05

Simplify the expressions

Simplify the expressions for sine and cosine of 75 degrees by combining the constants: \(\sin 75^{\circ}=\frac{\sqrt{3}+1}{2\sqrt{2}}\) \(\cos 75^{\circ}=\frac{\sqrt{3}-1}{2\sqrt{2}}\) The sine of 75 degrees is \(\frac{\sqrt{3}+1}{2\sqrt{2}}\), and the cosine of 75 degrees is \(\frac{\sqrt{3}-1}{2\sqrt{2}}\).

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