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Chapter 4: Problem 65

Describe a relationship among trigonometric functions that is based on angles that are complements.

Short Answer

Expert verified
The relationship among trigonometric functions based on complementary angles is given by the identities \( \sin(90^\circ - A) = \cos A \) and \( \cos(90^\circ - A) = \sin A \) for any angle A. This is due to the sine of one angle being equal to the cosine of its complement and vice versa.

Step by step solution

01

Define Complementary Angles

Two angles are said to be complementary if the sum of their measures equals 90 degrees. We use this quality to show the relationship between sine and cosine.
02

State the Relationship

The main relationship between trigonometric functions based on angles that are complements is given by the identities \( \sin(90^\circ - A) = \cos A \) and \( \cos(90^\circ - A) = \sin A \) for any angle A. This relationship is also valid in radians, where 90 degrees corresponds to \( \frac{\pi}{2} \) radians: \( \sin(\frac{\pi}{2} - A) = \cos A \) and \( \cos(\frac{\pi}{2} - A) = \sin A \).
03

Explain the Relationship

If two angles are complementary, the sine of one angle will be equal to the cosine of the other, and vice versa. This is due to the rotational symmetry of the unit circle definition of sine and cosine, where the sine represents the y-coordinate and the cosine represents the x-coordinate. If we rotate by \( \frac{\pi}{2} \) (or 90 degrees) radians, the x and y coordinates are swapped, which gives us this relationship.

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Most popular questions from this chapter

Let \(f(x)=\left\\{\begin{array}{ll}x^{2}+2 x-1 & \text { if } x \geq 2 \\ 3 x+1 & \text { if } x<2\end{array}\right.\) Find \(f(5)-f(-5) . \text { (Section } 1.3, \text { Example } 6)\)

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 \(g\) $$f(x)=\sin x, g(x)=\cos 2 x, h(x)=(f-g)(x)$$

Use a vertical shift to graph one period of the function. $$y=2 \sin \frac{1}{2} x+1$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-4 \cos \left(2 x-\frac{\pi}{2}\right)$$

Graph one period of each function. $$y=\left|3 \cos \frac{2 x}{3}\right|$$
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