Chapter 6
Inverse Trig Functions and Equations
Section 6.1 Inverse Trig Functions
Section 6.2 Trig Equations I
Section 6.3 Trig Equations II
Section 6.4 Equations
Section 6.1 Inverse Trig Functions
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Know and use the inverse sine
Know and use the inverse cosine
Know and use the inverse tangent
Know and use the inverse secant
Know and use the inverse cosecant
Know and use the inverse cotangent
Know and use inverse function values
Inverse Function
• The inverse function of the one-to-one
function f is defined as:
f -1 = {(y,x) | (x,y) belongs to f }
General Statements about Inverses
1. If the point (a,b) lies on the graph of the
one-to-one function f, then the point (b,a)
lies on the graph of f -1.
2. The domain of f is equal to the range of
f -1, and the range of f is equal to the
domain of f -1.
3. For all x in the domain of f , f -1[(x)] = x,
and all x in the domain of f -1, f [(x)]=x.
General Statements about Inverses
4. Because point (b, a) is a reflection of the
point (a, b) across the line y=x, the graph of
f -1 is the reflection of the graph of f across
this line.
5. To find the equation that defines the inverse
of a one-to-one function f , follow these
steps:
1.
2.
3.
Let y = f (x)
Interchange x and y in the equation
Solve for y, and then write y= f -1(x).
-1
sin x
or arcsin x
y = sin-1 x or y = arcsin x
means x=sin y, for y in [-é/2, é/2]
-1
cos x
or arccos x
y = cos-1 x or y = arccos x
means x=cos y, for y in [0, é]
-1
tan x
or arctan x
y = tan-1 x or y = arctan x
means x=tan y, for y in (-é/2, é/2)
-1
sec x
or arcsec x
y = sec-1 x or y = arcsec x
means y = cos-1 (1/x),
where x is in (-ë, -1]  [1, ë)
-1
csc x
or arccsc x
y = csc-1 x or y = arccsc x
means y = sin-1 (1/x),
where x is in (-ë, -1]  [1, ë)
-1
cot x
or arccot x
y = cot-1 x or y = arccot x means
y = tan-1 (1/x),if x is in [0, ë)
or
y = tan-1 (1/x) + é ,if x is in(-ë, 0)
Section 6.2 Trig Equations I
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Solve equations by linear methods
Solve equations by factoring
Solve equations by quadratic formula
Apply trig to music
Intersection of Graphs Method
• Graph each equation y=f(x) and y=g(x)
• Find the point(s) of intersection
x-Intercept Method
• Graph y=f(x)
• Find the x-Intercepts
• These are the solutions for f(x) = 0
Solving Trigonometric Equations
Algebraically
1. Decide whether the equation is linear or
quadratic (to determine the method)
2. If there is only on trig function, solve the
equation for that function
3. If there are more than one, rearrange the
equation so that one side equals 0. Try to
factor and set each factor = 0 to solve.
4. If #3 doesn’t work try using an identity or
square both sides.
5. If the equation is quadratic and not
factorable use the quadratic formula.
Solving Trig Equations
Graphically
For equations of the form f(x) = g(x):
1. Graph y=f(x) and y =g(x) over the
required domain.
2. Find the x-coordinates of the points of
intersection of the graphs.
Solving Trig Equations
Graphically (cont)
For equations of the form f(x) = 0:
1. Graph y=f(x) over the required domain.
2. Find the x-intercepts of the graphs.
Section 6.3 Trig Equations II
• Solve equations with half-angles
• Solve equations with multiple angles
• Apply trig to music
Section 6.4 Equations Involving
Inverse Trig Functions
• Solve for x in terms of y using inverse
functions
• Solve inverse trigonometric equations