91Ó°ÊÓ

Amplitude refers to the maximum value or height that a sine or cosine function can reach. In mathematical terms, it is the coefficient that comes before the sine or cosine function in an expression.

For example, in the function \[ y = A \sin(x + C) \], the value of \ A \ represents the amplitude. It tells us how high or low the wave goes from its central axis.

Calculating the amplitude is straightforward. You can use the formula: \[ A = \sqrt{a^2 + b^2} \] where \ a \ and \ b \ are the coefficients of \ \sin(x) \ and \ \cos(x) \ respectively. This formula gives you the exact maximum height the wave can achieve.

Let's apply this to our original example problem, where the coefficients of \ \sin(x) \ and \ \cos(x) \ are -\frac{\sqrt{3}}{2} \ and -\frac{1}{2}, respectively. Plugging these into our formula, we get:
\[ A = \sqrt{ \left( -\frac{\sqrt{3}}{2} \right)^2 + \left( -\frac{1}{2} \right)^2 } = \sqrt{ \frac{3}{4} + \frac{1}{4} } = \sqrt{1} = 1 \]
This tells us that the amplitude is 1.

Phase shift refers to the horizontal shift of the sine or cosine function on a graph. It shows how much the graph of the function is shifted from its usual position.

To find the phase shift, we use the formula:
\[ \sin C = \frac{b}{A} \text{ and } \cos C = \frac{a}{A} \]

Here, \ a \ and \ b \ are the coefficients of the sine and cosine terms, and \ A \ is the amplitude. Once you have \sin(C) \ and \cos(C), you can determine \ C \ using trigonometric identities.

In our example, \sin C \ and \cos C \ are determined as follows:
\[ \sin C = \frac{ -\frac{1}{2} }{1} = -\frac{1}{2} \text{ and } \cos C = \frac{ -\frac{\sqrt{3}}{2} }{1} = -\frac{\sqrt{3}}{2} \]

Knowing these values, you can recognize that they correspond to \ C = -\frac{5\pi}{6} \, as \ \sin( -\frac{5\pi}{6} ) = -\frac{1}{2} \ and \ \cos( -\frac{5\pi}{6} ) = -\frac{\sqrt{3}}{2} \.

Sine and cosine functions are fundamental in trigonometry. They help describe periodic phenomena like sound waves, light waves, or the motion of pendulums.

A standard sine or cosine function takes the form: \[ y = A \sin(x + C) \ or \ y = A \cos(x + C) \]

Here,

• \ A \ is the amplitude, indicating the height of the wave.
• \ x \ is the variable.
• \ C \ is the phase shift, indicating where the wave starts.

In our exercise, we need to reformat the expression \ -\frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x \ into the form \ A \sin (x + C) \. Applying the step-by-step solution, we find:

\ -\frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x = \sin \left( x - \-\frac{5\pi}{6} \right) \ which is a sine function shifted horizontally by \-\frac{5\pi}{6} \ units.
Understanding how to convert expressions this way helps to visualize and graph these functions better.