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Chapter 10: Problem 10

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \left(\frac{13 \pi}{12}\right)$$

Short Answer

Expert verified
\(- \frac{\sqrt{2+\sqrt{3}}}{2}\).

Step by step solution

01

Break Down the Angle

Notice that the angle \( \frac{13\pi}{12} \) can be rewritten as the sum of two angles whose cosine values we know. Let鈥檚 express \( \frac{13\pi}{12} \) as the sum of \( \pi \) and \( \frac{\pi}{12} \). Thus, \( \frac{13\pi}{12} = \pi + \frac{\pi}{12} \).
02

Apply the Cosine Addition Formula

Using the angle addition identity, \( \cos(a + b) = \cos a \cos b - \sin a \sin b \), substitute \( a = \pi \) and \( b = \frac{\pi}{12} \). Therefore, \( \cos(\pi + \frac{\pi}{12}) = \cos \pi \cos \frac{\pi}{12} - \sin \pi \sin \frac{\pi}{12} \).
03

Use Known Trigonometric Values

We know that \( \cos \pi = -1 \), \( \sin \pi = 0 \). Thus, \( \cos(\pi + \frac{\pi}{12}) = (-1) \cos \frac{\pi}{12} - 0 \cdot \sin \frac{\pi}{12} = -\cos \frac{\pi}{12} \). Now, focus on finding \( \cos \frac{\pi}{12} \).
04

Use Half-Angle Identity

To find \( \cos \frac{\pi}{12} \), let's use the half-angle identity: \( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}} \). We choose \( \theta = \frac{\pi}{6} \) so that \( \frac{\pi}{12} = \frac{\pi}{6} \times \frac{1}{2} \). Then \( \cos \frac{\pi}{12} = \sqrt{\frac{1 + \cos \frac{\pi}{6}}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \).
05

Simplify the Expression

Simplify the expression for \( \cos \frac{\pi}{12} \): \( \cos \frac{\pi}{12} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2+\sqrt{3}}}{2} \). Substitute this value back to find \( \cos \left( \frac{13\pi}{12} \right) = -\cos \frac{\pi}{12} = - \frac{\sqrt{2+\sqrt{3}}}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Addition Formula
The cosine addition formula is a powerful tool in trigonometry. It helps us find the cosine of the sum of two angles. The formula is expressed mathematically as:
  • \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)
This identity is useful for manipulating trigonometric expressions and finding precise values without a calculator.
In the given problem, this formula was used to break down the angle \(\frac{13\pi}{12}\) into more familiar angles, \(\pi\) and \(\frac{\pi}{12}\).
The values \(\cos \pi = -1\) and \(\sin \pi = 0\) were then substituted into the formula to simplify further. By handling the addition of angles this way, we could move closer to finding an exact expression for \(\cos\left(\frac{13\pi}{12}\right)\). Understanding this formula opens up the way to approximating and understanding more complicated angle measures in trigonometry.
Half-Angle Identity
The half-angle identity is a part of a set of identities in trigonometry that helps in finding the exact values for angles that aren't directly found on typical trigonometric charts. The half-angle identity for cosine is given by:
  • \(\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}}\)
This formula can produce two results, represented by the \(\pm\) sign. The sign is chosen based on the quadrant in which the angle \(\frac{\theta}{2}\) lies.
In this problem, finding \(\cos \frac{\pi}{12}\) involved applying the identity with \(\theta = \frac{\pi}{6}\), yielding the expression \(\sqrt{\frac{2 + \sqrt{3}}{4}}\).
This half-angle identity thus aids in breaking down the trigonometric function into more manageable parts, thereby assisting in finding exact trigonometric values for angles like \(\frac{\pi}{12}\) which aren't typically found on standard unit circle measurements.
Trigonometric Values
Trigonometric values for specific angles are a fundamental aspect of trigonometry. Knowing the exact values for angles such as \(0\), \(30\), \(45\), \(60\), and \(90\) degrees (or their radian equivalents) is crucial.
In this problem, recognizing that \(\cos \pi = -1\) and \(\sin \pi = 0\) allows simplification before applying additional identities.
The trigonometric values for certain angles are frequently used to either directly solve problems or to work through formulas like the cosine addition formula or the half-angle identity. By being familiar with these basic values, students can tackle harder trigonometry problems using identities, as demonstrated in the solution. Learning these values by heart enables quicker calculation and deeper understanding of angle measures and their trigonometric functions.

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$\cos \frac{\theta}{2}=1$$

Use a graphical method to solve each equation over the interval \([0,2 \pi) .\) Round values to the nearest thousandth. $$\sin 3 x-\sin x=0$$

Give the exact real number value of each expression. Do not use a calculator. $$\cos \left(\sin ^{-1} \frac{3}{5}-\cos ^{-1} \frac{12}{13}\right)$$

Verify that each equation is an identity by using any of the identities introduced in the first three sections of this chapter. $$\frac{1-\sin t}{\cos t}=\frac{1}{\sec t+\tan t}$$

Solve each problem. Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of the slight variation in the frequency. This phenomenon can be seen on a graphing calculator. (a) Consider two tones with frequencies of 220 and \(223 \mathrm{Hz}\) and pressures \(P_{1}=0.005 \sin 440 \pi t \quad\) and \(\quad P_{2}=0.005 \sin 446 \pi t\) respectively. A graph of \(P_{1}+P_{2}\) as \(Y_{3}\) felt by an eardrum over the 1 -second interval \([0.15,1.15]\) is shown here. How many beats are there in 1 second? (Graph can't copy) (b) Repeat part (a) with frequencies of 220 and 216 (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given.
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