13-4 The Sine Function Hubarth Algebra II

The sine function, 𝑦= sin 𝜃, matches the measure 𝜃 of an angle in standard position with the Y-coordinate of a point on the unit circle. This point is where the terminal side of the angle Intersects the unit circle.

Ex. 1 Interpreting the Sine Function in Degrees Use the graph of the sine function. a. What is the value of y = sin for = 180°? The value of the function at = 180° is 0. b. For what other value(s) of from 0° to 360° does the graph of sin have the same value as for = 180°? When y = 0, = 0° and 360°.

Ex. 2 Estimating Sine Values in Radians 𝜋 2 𝜋 2𝜋 3𝜋 2 Ex. 2 Estimating Sine Values in Radians Estimate each value from the graph. Check your estimate with a calculator. b. sin 2 a. sin 3 The sine function reaches its median value of 0 at 𝜋≈ 3.14. The value of the function at 3 is slightly more than 0, or about 0.1. The sine function reaches its maximum value of 1 at 𝜋 2 , so sin 𝜋 2 = 1. sin = 1 2 sin 3 = 0.1411200081  

Ex 3 Finding the Period of a Sine Curve Use the graph of y = sin 6 , for 0 ≤ ≤ 2 . a. How many cycles occur in this graph? How is the number of cycles related to the coefficient of in the equation? The graph shows 6 cycles. The number of cycles is equal to the coefficient of . b. Find the period of y = sin 6. 2 ÷ 6 = Divide the domain of the graph by the number of cycles. 3 The period of y = sin 6 is . 3

Ex. 4 Finding the Amplitude of a Sine Curve This graph shows the graph of y = a • sin𝜃 for values of a = 3 4 and a = 3. a. Find the amplitude of each sine curve. How does the value of a affect the amplitude? The amplitude of y = sin is 1, and the amplitude of y = • sin is . 3 4 The amplitude of y = 3 • sin is 3. In each case, the amplitude of the curve is | a |. b. How would a negative value of a affect each graph? When a is negative, the graph is a reflection in the x-axis.

Ex. 5 Sketching a Graph a. Sketch one cycle of a sine curve with amplitude 3 and period 4. Step 3: Since the amplitude is 3, the maximum 3 and the minimum is –3. Plot the five points and sketch the curve. Step 2: Mark equal spaces through one cycle by dividing the period into fourths. Step 1: Choose scales for the y-axis and the x-axis that are about equal ( 𝜋= 1 unit). On the x-axis, mark one period (4 units).

The amplitude is 3, and a > 0, so a = 3. b. Use the form y = a sin b . Write an equation with a > 0 for the sine curve in part a. The amplitude is 3, and a > 0, so a = 3. The period is 4, and 4 = , so b = . 2 b An equation for the function is y = 3 sin x. 2

Ex. 6 Graphing From a Function Rule Sketch one cycle of y = 5 3 sin 3𝜃 . | a | = , so the amplitude is . 5 3 Divide the period into fourths. Using the values of the amplitude and period, plot the zero-max-zero-min-zero pattern. Sketch the curve. b = 3, so there are 3 cycles from 0 to 2 . = , so the period is . 2 b 3

Ex. 7 Real-World Connection Find the period of the following sine curve. Then write an equation for the curve. According to the graph, one cycle takes 3 units to complete, so the period is 3. period = Use the relationship between the period and b. 2 b 3 = Substitute. b = Multiply each side by . 3 Use the form y = a sin b . An equation for the graph is y = 5 sin . 2 3