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Chapter 4: Problem 27

Find a cofunction with the same value as the given expression. $$\cos \frac{2 \pi}{5}$$

Short Answer

Expert verified
The cofunction with the same value as the given expression \(\cos \frac{2 \pi}{5}\) is \(\sin \frac{\pi}{10}\).

Step by step solution

01

Identity Recall

Recall the definition of a cofunction: for a given angle, the trigonometric functions sine and cosine are cofunctions. This means they yield the same value for complementary angles. In other words, \(\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right)\). Here, \(\theta = \frac{2 \pi}{5}\).
02

Calculate Complementary Angle

Subtract the given angle from \(\frac{\pi}{2}\) to get the complementary angle.\(\frac{\pi}{2}\) - \(\frac{2 \pi}{5} = \frac{5 \pi - 4 \pi}{10} = \frac{\pi}{10}\)
03

Express as Cofunction

Replace \(\theta\) by the complementary angle in the sine function to get the equivalent cofunction. So, \(\sin \frac{\pi}{10} = \cos \frac{2 \pi}{5}\).

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Most popular questions from this chapter

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-2.5 \sin \frac{\pi}{3} x \text { and } y=-2.5 \csc \frac{\pi}{3} x$$

The number of hours of daylight in Boston is given by $$ y=3 \sin \frac{2 \pi}{365}(x-79)+12 $$ where \(x\) is the number of days after January 1 a. What is the amplitude of this function? b. What is the period of this function? c. How many hours of daylight are there on the longest day of the year? d. How many hours of daylight are there on the shortest day of the year? e. Graph the function for one period, starting on January 1

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\tan 4 x$$

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=-2 \sin x, g(x)=\sin 2 x, h(x)=(f+g)(x)$$

Solve: \(\log _{3}(x+5)=2\) (Section 3.4, Example 6)
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