1.6 Trigonometric Functions AP Calculus AB/BC 1.6 Trigonometric Functions

Trigonometric functions are used extensively in calculus. When you use trig functions in calculus, you must use radian measure for the angles. The best plan is to set the calculator mode to radians. Radians can be converted to degrees by multiplying the radian measure by Now, let’s quickly review the six trig functions. The definitions are in your book on page 46.

The Six Trigonometric Functions r is the radius of the circle formed through the point (x, y) with center at O. Also, r is always a nonzero positive number. r y x The Basic Functions The Reciprocal Functions

Sine and Cosine

Tangent and Cotangent Remember, the cotangent is the reciprocal of the tangent.

Secant and Cosecant Remember, the secant is the reciprocal of the cosine. Remember, the cosecant is the reciprocal of the sine.

Definition: Periodic Function and Period A function f(x) is periodic if there is a positive number p such that f(x + p) = f(x) for every value of x. The smallest such value of p is the period of f. In other words, periodic functions repeat themselves, and the period is the length of one cycle of the function.

Even and Odd Trig Functions: “Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change. Cosine is an even function because: Secant is also an even function, because it is the reciprocal of cosine. Even functions are symmetric about the y - axis.

Even and Odd Trig Functions: “Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the sign of the y value also changes. Sine is an odd function because: Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function. Odd functions have origin symmetry.

Example: Find all the trigonometric values of θ with the given condition: By definition, x = −15 r = 17 to solve for the missing piece. Since sin θ > 0, y = 8 (not −8) Also note the angle is in the 2nd quadrant since sin θ > 0 and cos θ < 0. Now, use the trig definitions to find the remaining values.

The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions. Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. is a stretch. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. is a shrink. The horizontal changes happen in the opposite direction to what you might expect.

When we apply these rules to sine and cosine, we use some different terms. is the amplitude. Vertical shift is the period. Horizontal shift B A C D

Example: Temperatures in Fairbanks, AK Example: Temperatures in Fairbanks, AK. Find the amplitude, period, horizontal shift, vertical shift, domain, and range of the model Amplitude = 37 Domain = (−∞, ∞) Range = [−A + D, A + D] Period = 365 Range = [−37 + 25, 37 + 25] Horizontal Shift = 101 Range = [−12, 62] Vertical Shift = 25

The sine equation is built into the TI-84 as a sinusoidal regression equation. For practice, we will find the sinusoidal equation for the tuning fork data in the book. To save time, we will use only five points instead of all the data.

Tuning Fork Data Time: .00108 .00198 .00289 .00379 .00471 Pressure: .200 .771 -.309 .480 .581 2nd { .00108, .00198, .00289, .00379, .00471 2nd } STO 2nd L 1 ENTER ENTER STAT CALC The calculator should return:

The calculator gives you an equation and constants: ENTER STAT CALC The calculator should return: The calculator gives you an equation and constants:

We can use the calculator to plot the new curve along with the original points: Y= y1=regeq(x) VARS 5:Statistics EQ 1:RegEQ Use to highlight Plot 1 ENTER WINDOW

WINDOW GRAPH

Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. * Stop here, or continue to include regression functions on the TI-89. These restricted trig functions have inverses. Inverse trig functions and their restricted domains and ranges are defined in the book on page 50.

Inverse Trigonometric Functions

Example: Solve for x. Plug this into your calculator. x = −1.107 This solution is in the fourth quadrant which is good since the interval which we are interested in is from Since the period of tan is π, the solution is x = −1.107 + kπ. This gives all the solutions for the value of x whereas the inverse function only gave one value for x.

p Now, it’s time for you to try a problem. (No peaking ahead!) Evaluate the expression p