Trig Functions

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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.
2nd o
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 xaxis
Horizontal stretch or shrink;reflection about yaxis
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 onetoone.
However, the domain can be restricted for trig functions to make them onetoone.
These restricted trig functions have inverses.
Inverse trig functions and their restricted domains and ranges are defined in the book.
siny x