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Chapter 5: Problem 89

Without drawing a graph, describe the behavior of the basic sine curve.

Short Answer

Expert verified
The sine function oscillates between -1 and 1 with a period of \( 2\pi \) radians or 360 degrees and has no phase shift.

Step by step solution

01

Initial Value and Amplitude of Sine Function

The sine function, denoted by sin(x), is an oscillatory function. It starts at 0, increases till \( \frac{\pi}{2} \), then decreases till \( \pi \), again increases till \( \frac{3\pi}{2} \), and finally decreases and come back to 0 at \( 2\pi \). The maximum value, or the amplitude, of a sine function is 1 while the minimum is -1.
02

Period of Sine Function

The period of sine function is \( 2\pi \). This means it completes a full cycle within \( 2\pi \) radians or 360 degrees.
03

Phase Shift of Sine Function

Since the sine function starts from 0, it's phase shift is zero. Phase shift refers to the horizontal shift of the function.

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Most popular questions from this chapter

Prove that if \(x>0, \tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\frac{\pi}{2}\).

Use a graphing utility to graph two periodsof the function. Use a graphing utility to graph Use a graphing utility to graph \( y=\sin x-\frac{\sin 3 x}{9}+\frac{\sin 5 x}{25} \) in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by \([-2,2,1]\) viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?

Use a graphing utility to graph two periodsof the function. Use a graphing utility to graph \(y=\sin x+\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+\frac{\sin 4 x}{4}\) in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by \([-2,2,1]\) viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?

Determine the domain and the range of each function. $$ f(x)=\sin \left(\sin ^{-1} x\right) $$

Explain the difference between positive and negative angles. What are coterminal angles?
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