Warm UP! – Draw these 2 graphs on your paper Assume that each graph is scaled the same: Label a wavelength in each of the graphs. Which graph has a higher frequency? Which graph has a larger amplitude?

LG 4-2 Circular Functions MA3A3. Students will investigate and use the graphs of the six trigonometric functions. a. Understand and apply the six basic trigonometric functions as functions of real numbers. b. Determine the characteristics of the graphs of the six basic trigonometric functions. c. Graph transformations of trigonometric functions including changing period, amplitude, phase shift, and vertical shift. d. Apply graphs of trigonometric functions in realistic contexts involving periodic phenomena. MA3A8. Students will investigate and use inverse sine, inverse cosine, and inverse tangent functions. a. Find values of the above functions using technology as appropriate. b. Determine characteristics of the above functions and their graphs.

Stresses in the earth compress rock formations and cause them to buckle into sinusoidal shapes. It is important for geologists to be able to predict the depth of a rock formation at a given point. Such information can be very useful for structural engineers as well. In this learning goal we’ll learn about the circular functions, which are closely related to the trigonometric functions. Geologists and engineers use these functions as mathematical models to perform calculations for such wavy rock formations.

What is a Circular Function? Trigonometric functions are defined so that their domains are sets of angles and their ranges are sets of real numbers. Circular functions are defined such that their domains are sets of numbers that correspond to the measures (in radian units) of the angles of analogous trigonometric functions. The ranges of these circular functions, like their analogous trigonometric functions, are sets of real numbers. These functions are called circular functions because radian measures of angles are determined by the lengths of arcs of circles. In particular, trigonometric functions defined using the unit circle lead directly to these circular functions.

Circular Function Overview Graphically – The independent variable is now x rather than Ѳ so that we can fit sinusoids to situations that do not involve angles. Algebraically – Particular Equation Circular functions are just like the trigonometric functions except that the independent variable is an arc of a unit circle instead of an angle. Angles in radians form the link between angles in degrees and numbers of units of arc length.

Circular Functions A child is riding a ferris wheel at the fair. At a specific time, the height of the ride (h in meters) as a function of the distance from the ground (x in meters) is modeled by the circular function: What will the height of the child be if she is 12 meters above the ground at this time?

Another Example The displacement in inches of the ground at a certain point seconds after an earthquake is modeled by the circular function: What will the height of the ground at this point be 15 seconds after the earthquake?

Propagating Waves Waves moving through space and time Examples: Light Sound Water They come in different sizes Mirror repetition of the circle

Sound Waves and Light Waves Musical Notes Hearing Test Light Waves

Propagating Waves Notes

Example 1 The period of a wave is 10sec. What is the frequency? The waves completes 0.1 cycles every second.

Example 2 The musical note “A” is a sound wave with frequency of 440 Hz. The wavelength is 77.3cm. What is the speed of the sound wave? 340 m/s is the approximate speed of all sound waves through the air (depends on temperature and pressure).

Example 3 (test question!) Write a sine equation that models a “B” note with frequency of 494Hz and intensity of 0.75. To find the value of ‘b’ for the transformation, always set and solve for ‘b.’

Classwork Complete the worksheet (front and back) Remember to convert units to meters if they are not already.

Circular Functions Plot the graph of y = 4 cos 5x on your calculator, in radian mode. Find the period graphically and algebraically. Compare your results with your neighbor. Your graph should look like this: Tracing the graph, you find that the first high point beyond x = 0 is between x = 1.25 and x = 1.3. So graphically the period is between 1.25 and 1.3. To find the period algebraically, recall that the 5 in the argument of the cosine function is the reciprocal of the horizontal dilation. The period of the parent cosine function is 2π, because there are 2π radians in a complete revolution. Thus the period of the given function is

Find a particular equation for this sinusoid function: Notice that the horizontal axis is labeled x, not Ѳ, indicating that the angle is measured in radians. Sinusoidal axis is at y = 3, so K = 3 Amplitude is 2, so A = 2 From one high point to the next is 11 - 1. Period is 10. Dilation is so B = . Phase displacement is 1 (for y = cos x), H = 1. Write the particular equation: y = 3 + 2 cos (x - 1) Plotting this equation in radian mode confirms that it is correct. y = A cos B(x - H) + K