Trig Functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008

Black Canyon of the GunnisonNational Park, Colorado

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.

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If you want to brush up on trig functions, they are graphed in your book.

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: cos cos

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: sin sin

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.

y a f b x c d

Vertical stretch or shrink;reflection about x-axis

Horizontal stretch or shrink;reflection about y-axis

Horizontal shift

Vertical shift

Positive c moves left.

Positive d moves up.

The horizontal changes happen in the opposite direction to what you might expect.

is a stretch.1a

is a shrink.1b

When we apply these rules to sine and cosine, we use some different terms.

2sinf x A x C D

B

Horizontal shift

Vertical shift

is the amplitude.A

is the period.B

A

B

C

D 21.5sin 1 2

4y x

2 3

2

2

2

3

2

2

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 in the book.

siny x