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Chapter 4: Problem 89

In Exercises \(87-92\), find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. $$\sin \frac{11 \pi}{4} \cos \frac{5 \pi}{6}+\cos \frac{11 \pi}{4} \sin \frac{5 \pi}{6}$$

Short Answer

Expert verified
The first task is recognizing the addition formula for sine. Given that 'a' is \(\frac{11 \pi}{4}\) and 'b' is \(\frac{5 \pi}{6}\), by substitution, the given expression simplifies to \(\sin(\frac{11 \pi}{4}+\frac{5 \pi}{6})\). After simplifying the resulting radian measurement, use the unit circle to calculate the exact value.

Step by step solution

01

Identify the formula to be used

In this case, the addition formula for sine is being applied, which is given as \(\sin(a+b) = \sin a \cos b + \cos a \sin b\). The problem is to rearrange the given expression and identify 'a' and 'b' in this formula.
02

Identify 'a' and 'b'

Looking at the given expression, 'a' can be identified as \(\frac{11 \pi}{4}\) and 'b' can be identified as \(\frac{5 \pi}{6}\).
03

Substitute 'a' and 'b' into the formula

Substitute 'a' and 'b' into the formula \(\sin(a+b)\), which would result in \(\sin(\frac{11 \pi}{4}+\frac{5 \pi}{6})\)
04

Simplify the expression

After adding \(\frac{11 \pi}{4}\) and \(\frac{5 \pi}{6}\), you will have a radian measurement. Simplify this to find an equivalent measure in terms of \(2\pi\).
05

Calculate the sine

Now that the radian measurement is within the interval [0, \(2\pi\)], the exact value of the sine function can be found

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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. After using the four-step procedure to graph \(y=-\cot \left(x+\frac{\pi}{4}\right),\) I checked my graph by verifying it was the graph of \(y=\cot x\) shifted left \(\frac{\pi}{4}\) unit and reflected about the \(x\) -axis.

Describe a general procedure for obtaining the graph of \(y=A \sin (B x-C)\)

Use a graphing utility to graph two periods of the function. $$y=-2 \cos \left(2 \pi x-\frac{\pi}{2}\right)$$

A clock with an hour hand that is 15 inches long is hanging on a wall. At noon, the distance between the tip of the hour hand and the ceiling is 23 inches. At 3 P.M., the distance is 38 inches; at 6 P.M., 53 inches; at 9 P.M., 38 inches; and at midnight the distance is again 23 inches. If \(y\) represents the distance between the tip of the hour hand and the ceiling \(x\) hours after noon, make a graph that displays the information for \(0 \leq x \leq 24\)

The average monthly temperature, \(y,\) in degrees Fahrenheit, for Juneau, Alaska, can be modeled by \(y=16 \sin \left(\frac{\pi}{6} x-\frac{2 \pi}{3}\right)+40,\) where \(x\) is the month of the year \(\quad\) (January \(=1,\) February \(=2, \ldots\) December \(=12\) ). Graph the function for \(1 \leq x \leq 12 .\) What is the highest average monthly temperature? In which month does this occur?
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