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Chapter 5: Problem 26

Find the exact values of the sine, cosine, and tangent of the angle. $$15^{\circ}$$

Short Answer

Expert verified
The values are \(\sin 15 = \frac{\sqrt6 - \sqrt2}{4}, \cos 15 = \frac{\sqrt6 + \sqrt2}{4},\) and \(\tan 15 = 2 - \sqrt3\)

Step by step solution

01

Express in terms of Standard Angles

Firstly, express 15 degrees as a difference of standard angles that we know the trigonometric values. The standard angles are \(0, 30, 45, 60,\) and \(90\) degrees. In this case, \(15\) degrees can be written as \(45 - 30\).
02

Apply Formulae

Next, apply the following formulae:\n\nFor sine, \(\sin(a - b) = \sin a \cos b - \cos a \sin b\).\nFor cosine, \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).\nFor tangent, \(\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\).
03

Compute values

Now, substitute \(a = 45\) degrees and \(b = 30\) degrees, and use standard values. For example, \(\sin 45 = \cos 45 = \frac{1}{\sqrt{2}}\) and \(\sin 30 = \frac{1}{2}\). Calculate the values of sine, cosine and tangent.

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Most popular questions from this chapter

(a) determine the quadrant in which \(u / 2\) lies, and (b) find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\cos u=7 / 25, \quad 0

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