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Chapter 5: Problem 59

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Short Answer

Expert verified
The exact values of the sine, cosine and tangent of \(75^{\circ}\) are \(\frac{\sqrt{6} + \sqrt{2}}{4}\), \(\frac{\sqrt{6} - \sqrt{2}}{4}\) and \(\sqrt{3} + 1\) respectively.

Step by step solution

01

Expressing the given angle

Express the given angle \(75^{\circ}\) as the sum of \(45^{\circ}\) and \(30^{\circ}\) i.e., \(75^{\circ} = 45^{\circ} + 30^{\circ}\)
02

Calculating the sine value

Using the formula for sine of sum of two angles \(\sin A + B = \sin A \cos B + \cos A \sin B\), we find \(\sin 75^{\circ} = \sin(45^{\circ} + 30^{\circ}) = \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} = \frac{1}{\sqrt{2}}*\frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}}*\frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\)
03

Calculating the cosine value

Using the formula for cosine of sum of two angles \(\cos A + B = \cos A \cos B - \sin A \sin B\), we find \(\cos 75^{\circ} = \cos(45^{\circ} + 30^{\circ}) = \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ} = \frac{1}{\sqrt{2}}*\frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}}*\frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}\)
04

Calculating the tangent value

The tangent of an angle is the ratio of its sine to its cosine. Hence, \(\tan 75^{\circ} = \frac{\sin 75^{\circ}}{\cos 75^{\circ}} = \frac{\frac{\sqrt{6} + \sqrt{2}}{4}}{\frac{\sqrt{6} - \sqrt{2}}{4}} = \sqrt{3} + 1\)

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