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Chapter 5: Problem 22

Find the exact values of the sine, cosine, and tangent of the angle.$$\frac{5 \pi}{12}$$

Short Answer

Expert verified
The exact values of the sine, cosine, and tangent of the angle \(\frac{5\pi}{12}\) are \(\sin\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}\), \(\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}\), and \(\tan\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\), respectively.

Step by step solution

01

Express the Given Angle as a Sum/Difference of Common Angles

Express the given angle \(\frac{5\pi}{12}\) as a sum or difference of \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\), as these angles have known trigonometric values. This expression could be: \[\frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}\]
02

Find the Sine Value

Substitute the expression from above into the sine addition formula: \[\sin(a + b) = \sin a \cos b + \cos a \sin b\]Using the equation above, and noticing that \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), \(\sin \frac{\pi}{6} = \frac{1}{2}\), \(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), and \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), we get:\[\sin\left(\frac{5\pi}{12}\right) = \sin \frac{\pi}{4} \cos \frac{\pi}{6} + \cos \frac{\pi}{4} \sin \frac{\pi}{6} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\]
03

Find the Cosine Value

We now use the cosine addition formula: \[\cos(a + b) = \cos a \cos b - \sin a \sin b\]This gives:\[\cos\left(\frac{5\pi}{12}\right) = \cos \frac{\pi}{4} \cos \frac{\pi}{6} - \sin \frac{\pi}{4} \sin \frac{\pi}{6} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}\]
04

Find the Tangent Value

The tangent of an angle is simply the ratio of the sine to the cosine of the angle. Therefore, we have:\[\tan\left(\frac{5\pi}{12}\right) = \frac{\sin\left(\frac{5\pi}{12}\right)}{\cos\left(\frac{5\pi}{12}\right)} = \frac{\frac{\sqrt{6} + \sqrt{2}}{4}}{\frac{\sqrt{6} - \sqrt{2}}{4}} = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\]

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Key Concepts

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

Sine Addition Formula
Understanding the sine addition formula is valuable when evaluating trigonometric functions of sums of angles, especially when those angles are not directly available on the unit circle. The sine addition formula states that for any two angles, the sine of their sum can be expressed as:

\[ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \]
In practice, this formula allows us to break down complex angles into known components. For example, if an angle can be expressed as a sum of special angles like \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\), we can use their known sine and cosine values to find the sine of the original angle. This step-by-step method simplifies the process of finding exact trigonometric values for uncommon angles.
Cosine Addition Formula
Similar to the sine addition formula, the cosine addition formula plays a crucial role in computing the cosine of a sum of angles. The formula is expressed as:

\[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \]
This formula indicates that by knowing the cosine and sine values of individual angles, you can calculate the cosine value of their sum. It's essential for finding exact values of cosines for angles that may not appear directly on the standard unit circle. For example, when you have an angle like \(\frac{5\pi}{12}\), which is the sum of \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\), you can apply the formula with their known trigonometric values to obtain a precise result.
Tangent of an Angle
The tangent of an angle in trigonometry is a ratio that represents the slope of a line. This ratio is the sine of the angle divided by the cosine of the same angle. Mathematically, it is given by:

\[ \tan(a) = \frac{\sin(a)}{\cos(a)} \]
To find the tangent of an angle derived from the sum of two different angles, one can use the results from the sine and cosine addition formulas. After computing the sine and cosine separately, their quotient will yield the exact value for the tangent of the compound angle. This approach is incredibly helpful for angles like \(\frac{5\pi}{12}\) that don't have their tangent values readily memorized or easily accessible. By leveraging this method, we can determine the tangent of complex angles with precision.

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Most popular questions from this chapter

Solving a Trigonometric Equation In Exercises \(69-74,\) find all solutions of the equation in the interval \([0,2 \pi)\).$$\tan (x+\pi)+2 \sin (x+\pi)=0$$

Find the exact value of the trigonometric expression given that \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} .\) (Both \(u\) and \(v\) are in Quadrant II.)$$\sec (v-u)$$.

Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing Window. Use the graphs to determine whether \(y_{1}=y_{2}\).Explain your reasoning.$$y_{1}=\cos (x+2), \quad y_{2}=\cos x+\cos 2$$.

Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\sin 3 \theta+\sin \theta$$

Solving a Trigonometric Equation In Exercises \(69-74,\) find all solutions of the equation in the interval \([0,2 \pi)\).$$\cos (x+\pi)-\cos x-1=0$$
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