1.6 Trig Functions Greg Kelly, Hanford High School, Richland, Washington

1.6 Trig Functions The Mean Streak, Cedar Point Amusement Park, Sandusky, OH

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 and use when you need to use degrees. 2nd o If you want to brush up on trig functions, they are graphed on page 41.

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.

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

The sine equation is built into the TI-89 as a sinusoidal regression equation. For practice, we will find the sinusoidal equation for the tuning fork data on page 45. 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 alpha L 1 ENTER 6 3 9 alpha L 1 , alpha L 2 ENTER 2nd MATH SinReg The calculator should return: Statistics Regressions Done

6 3 9 alpha L 1 , alpha L 2 ENTER 2nd MATH SinReg The calculator should return: Statistics Regressions Done 6 8 ENTER 2nd MATH Statistics ShowStat 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) x ) regeq 2nd VAR-LINK Plot 1 ENTER ENTER WINDOW

Plot 1 ENTER ENTER WINDOW GRAPH

WINDOW GRAPH You could use the “trace” function to investigate the pressure at any given time.

p Trig functions are not one-to-one. However, the domain can be restricted for trig functions to make them one-to-one. These restricted trig functions have inverses. Inverse trig functions and their restricted domains and ranges are defined on page 47. p