Section 5.3 Trigonometric Graphs
Graphs of the Sine and Cosine Functions
To help us graph the sine and cosine functions, we rst observe that these functions repeat their
values in a regular fashion. To see exactly how this happens, recall that the circumf
Section 5.5 Inverse Trigonometric Functions and Their
Graphs
DEFINITION: The inverse sine function, denoted by sin1 x (or arcsin x), is dened to be
the inverse of the restricted sine function
sin x, x
2
2
DEFINITION: The inverse cosine function, denoted
Section 5.4 More Trigonometric Graphs
Graphs of the Tangent, Cotangent, Secant, and Cosecant Function
1
REMARK: Many curves have a U shape near zero. For example, notice that the functions
sec x and x2 + 1 are very dierent functions. But when we casually
Section 7.5 More Trigonometric Equations
EXAMPLE: Solve the equation 1 + sin x = 2 cos2 x.
Solution: We have
1 + sin x = 2 cos2 x
1 + sin x = 2(1 sin2 x)
1 + sin x = 2 2 sin2 x
2 sin2 x + sin x 1 = 0
(2 sin x 1)(sin x + 1) = 0
2 sin x 1 = 0
or
sin x + 1 =
Section 5.1 The Unit Circle
The Unit Circle
EXAMPLE: Show that the point
36
,
33
is on the unit circle.
Solution: We need to show that this point satises the equation of the unit circle, that is,
x2 + y 2 = 1. Since
2
2
36
3
6
+
= + =1
3
3
99
P is on the
Section 7.1 Trigonometric Identities
EXAMPLE: Simplify the expression cos t + tan t sin t.
Solution: We have
cos t + tan t sin t = cos t +
sin t
cos t
sin t = cos t +
cos2 t sin2 t
sin2 t
=
+
cos t
cos t
cos t
=
EXAMPLE: Simplify the expression tan +
1
co
Section 7.2 Addition and Subtraction Formulas
EXAMPLE: Find the exact value of each expression.
(a) cos 75
(b) cos
12
Solution:
(a) Notice that 75 = 45 + 30 . Since we know the exact values of sine and cosine at45 and
30 , we use the addition formula for
Section 7.4 Trigonometric Equations
2
EXAMPLE: Solve the equation cos x =
, and list eight specic solutions.
2
Solution:
Find solutions in one period. Because cosine has period 2, we rst nd the solutions in
any interval of length 2. The solutions are
x=
Section 7.3 Double-Angle, Half-Angle, and Sum-Product
Identities
Double-Angle Formulas
2
and x is in quadrant II, nd cos 2x and sin 2x.
3
Solution: Using one of the double-angle formulas for cosine, we get
EXAMPLE: If cos x =
cos 2x = 2 cos2 x 1 = 2
2
3
Section 5.2 Trigonometric Functions of Real Numbers
The Trigonometric Functions
EXAMPLE: Use the Table below to nd the six trigonometric functions of each given real
number t.
(a) t =
(b) t =
3
2
1
EXAMPLE: Use the Table below to nd the six trigonometric
Section 8.1 Polar Coordinates
Denition of Polar Coordinates
DEFINITION: The polar coordinate system is a two-dimensional coordinate system in which each
point P on a plane is determined by a distance r from a xed point O that is called the pole (or origin
Section 8.2 Graphs of Polar Equations
Graphing Polar Equations
The graph of a polar equation r = f (), or more generally F (r, ) = 0, consists of all points P that
have at least one polar representation (r, ) whose coordinates satisfy the equation.
EXAMPL
Section 8.4 Plane Curves and Parametric Equations
Suppose that x and y are both given as functions of a third variable t (called a parameter) by the
equations
x = f (t),
y = g (t)
(called parametric equations). Each value of t determines a point (x, y ),