13.4 – 13.4 – The Sine The Sine FunctionFunction

I. I. Interpreting Sine Interpreting Sine FunctionsFunctions The sine function, y = sin The sine function, y = sin θθ, matches , matches

the measure the measure θθ of an angle in standard of an angle in standard position with the y – coordinate on the position with the y – coordinate on the unit circle.unit circle.

The periodic sine function has one The periodic sine function has one cycle every 360° or 2cycle every 360° or 2ππ

– Hence the sine of any degree greater than Hence the sine of any degree greater than 22ππ, then , then the cycle is repeated.the cycle is repeated.

The x-axis or domain will be thought The x-axis or domain will be thought of in terms of of in terms of theta, theta, θθ

When graphing you have to pay When graphing you have to pay attention to the domain of attention to the domain of theta. theta. Trig Cycles are repeated once every Trig Cycles are repeated once every 360 degrees or 2360 degrees or 2ππ

The General Sine Curve:The General Sine Curve:

Example 1: Graph, using a table the Example 1: Graph, using a table the values of the sine curve in the values of the sine curve in the domain domain

0 0 ≤ ≤ θθ ≤ 2 ≤ 2ππ. Use only the coordinates . Use only the coordinates on the on the θθ and y axes. and y axes.

You can vary the period and You can vary the period and amplitude of the sine curve to get amplitude of the sine curve to get different curves, either “extended” or different curves, either “extended” or more “frequent”more “frequent”

Done my multiplying the function by a Done my multiplying the function by a constant, and constant, and thetatheta by a constant by a constant

Y = 3 sin Y = 3 sin θθ

Y = sin 2Y = sin 2θθ

II. II. Properties of the Sine Properties of the Sine FunctionFunction y = y = a sin ba sin bθθ

ІІaaІІ = the amplitude (the highest and lowest = the amplitude (the highest and lowest points on the curves.) points on the curves.)

b = the number of cycles the curve makes b = the number of cycles the curve makes from 0 to 2from 0 to 2ππ

22ππ / b = the period of the cycle / b = the period of the cycle

Steps to Graphing y = Steps to Graphing y = a sin ba sin bθθ

– Step 1: draw your reference curveStep 1: draw your reference curve– Step 2: identify the domain and set up your Step 2: identify the domain and set up your

graph.graph.– Step 3: identify the amplitude and documentStep 3: identify the amplitude and document– Step 4: identify the number of cycles in the Step 4: identify the number of cycles in the

domain that you have to graphdomain that you have to graph– Step 4: identify the period you need to create Step 4: identify the period you need to create

one cycleone cycle– Step 5: Alter graph in needed in increments of Step 5: Alter graph in needed in increments of

4 units on the theta axis to meet the criteria of 4 units on the theta axis to meet the criteria of the total cycles.the total cycles.

Example 2: how many cycles does Example 2: how many cycles does the following have in the domain the following have in the domain from 0 to 2from 0 to 2ππ? Give the amplitude and ? Give the amplitude and period as well.period as well.

A) y = -5 sin 2A) y = -5 sin 2θθ

B) f(x) = .5 sin .5B) f(x) = .5 sin .5θθ

Example 3: Graph the following. Example 3: Graph the following. (check on the calculator)(check on the calculator)

A) y = -2 sin 2A) y = -2 sin 2θθ, for 0 < , for 0 < θθ < 2 < 2ππ B) y = 5 sin B) y = 5 sin θθ, for 0 < , for 0 < θθ < 2 < 2ππ C) y = ½ sin .5C) y = ½ sin .5θθ, for 0 < , for 0 < θθ < 2 < 2ππ