Problem 89 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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.