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Chapter 5: Problem 20

Find the exact value of each expression. $$\cos 105^{\circ}$$

Short Answer

Expert verified
\(\cos(105^{\circ}) = (\sqrt{2} - \sqrt{6})/4\).

Step by step solution

01

Express 105 degrees in terms of standard angles

105 degrees can be expressed as the sum of 60 degrees and 45 degrees, both of which are standard angles in the unit circle. So, we have \(105^{\circ}\) = \(60^{\circ} + 45^{\circ}\)
02

Apply the formula for cosine of a sum

Using the trigonometric identity for the cosine of a sum of two angles, we get: \(\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)\). Substituting \(a = 60^{\circ}\) and \(b = 45^{\circ}\), we get: \(\cos(105^{\circ}) = \cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ})\).
03

Calculate the final result

We know that \(\cos(60^{\circ}) = 1/2\), \(\cos(45^{\circ}) = \sqrt{2}/2\), \(\sin(60^{\circ}) = \sqrt{3}/2\), and \(\sin(45^{\circ}) = \sqrt{2}/2\). So, substituting these values into the equation, we find: \(\cos(105^{\circ}) = (1/2) * (\sqrt{2}/2) - (\sqrt{3}/2) * (\sqrt{2}/2) = \sqrt{2}/4 - \sqrt{6}/4 = (\sqrt{2} - \sqrt{6})/4\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. To prove a trigonometric identity, I select one side of the equation and transform it until it is the other side of the equation, or I manipulate both sides to a common trigonometric expression.

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$2 \tan ^{2} x+5 \tan x+3=0$$

Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\sin x+\sin 2 x+\sin 3 x=0$$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$5 \cot ^{2} x-15=0$$

Determine the amplitude and period of \(y=3 \sin \frac{1}{2} x\) Then graph the function for \(0 \leq x \leq 4 \pi\) (Section 4.5, Example 3)
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