Write y(t)=2sin4πt+5cos4πt in the form y(t)=asin(wt+φ) and identify the amplitude, angular frequency, and the phase shift of the spring motion. record your answers in the response box.

Angular frequency:

Phase shift:

Amplitude:

Step-by-step explanation:

Trigonometric Transformations

There's a number of trigonometric identities that can help us to transform expressions into others more useful or convenient. One well-known relation is the sine of the sum of angles:

We also know

We are given the expression

and are required to transform to the form

Where A,w and must be determined. We apply the formula for the sine of the sum of angles

This must be equal to the given expression

Being the phase shift and therefore independent of time, we can safely equate term to term

The time-dependant parts are equal, so

a)

Both relations give us the angular frequency w

Now the rest of the equations allow us to find \phi and A

Taking squares of both and adding them

Given that

We have

Dividing [2] by [1]

Or

Summarizing

Angular frequency:

Phase shift:

Amplitude:

the ratio of white tiles to blue tiles is 48: 16. the ratio in it's simplest form is 3: 1.

step-by-step explanation:

ratio: take the first number and add a colon next to it, then add the second number after that. make sure the ratio order matches up with the written problem!

simplest form: we can simplify the ratio 48 : 16 by dividing both terms by the greatest common factor (gcf).

the gcf of 48 and 16 is 16.

divide both terms by 16:

48 ÷ 16 = 3

16 ÷ 16 = 1

reducing to simplest form:

48 : 16 = 3 : 1

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0