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Chapter 5: Problem 33

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 the angle \(75^{\circ}\) are \(sin75^{\circ} =\frac{\sqrt{6}+\sqrt{2}}{4}\), \(cos75^{\circ} = \frac{\sqrt{6}-\sqrt{2}}{4}\), \( tan75^{\circ} = 2 + \sqrt{3}\) respectively.

Step by step solution

01

Express the non-standard angle as the sum of two standard angles

Let's write \(75^{\circ}\) as the sum of \(45^{\circ}\) and \(30^{\circ}\).
02

Calculation of cosine

Using cos addition formulas \(cos(A + B) = cosA*cosB - sinA*sinB\), we determine \(cos75^{\circ}\) by substituting \(A = 45^{\circ}\) and \(B = 30^{\circ}\). After calculation, we get \(cos75^{\circ} =\frac{\sqrt{6} - \sqrt{2}}{4}\).
03

Calculation of sine

We calculate the sine of \(75^{\circ}\) using Pythagorean identity that is \(sin 胃 = 卤 \sqrt{1 - cos^2 胃}\). Thus \(sin75^{\circ} =\sqrt {1- ((\sqrt6 - \sqrt2)^2)/16} =\sqrt {1- (6 - 2\sqrt3 + 2)/16} =\sqrt{(10 + 2\sqrt{3})/16}= \sqrt{(5 + \sqrt{3})/8} = \frac{\sqrt{6} + \sqrt{2}}{4}\).
04

Calculation of tangent

We calculate tangent of \(75^{\circ}\) using the formula \(tan 胃 = \frac{sin 胃}{cos 胃}\). Thus \(tan(75^{\circ}) = \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}-\sqrt{2}} = 2 + \sqrt{3}\).

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