Transcript

Section 6.1 Inverse Trig Functions

Section 6.2 Trig Equations I

Section 6.3 Trig Equations II

Section 6.4 Equations

Chapter 6Inverse Trig Functions and Equations

Section 6.1 Inverse Trig Functions

• 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 ff, then the point (b,a) lies on the graph of ff -1-1..

2. The domain of ff is equal to the range of ff -1-1, and the range of ff is equal to the domain of ff -1-1.

3. For all x in the domain of ff , ff -1-1[(x)] = x, and all x in the domain of ff -1-1, ff [(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 ff -1-1 is the reflection of the graph of ff across this line.

5. To find the equation that defines the inverse of a one-to-one function ff , follow these steps:

1. Let y = ff (x)2. Interchange x and y in the equation3. Solve for y, and then write y= f f -1-1(x).

sin-1x or arcsin x

y = sin-1 x or y = arcsin x

means x=sin y, for y in [-é/2, é/2]

cos-1x or arccos x

y = cos-1 x or y = arccos x

means x=cos y, for y in [0, é]

tan-1x or arctan x

y = tan-1 x or y = arctan x

means x=tan y, for y in (-é/2, é/2)

sec-1x 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)

csc-1x 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)

cot-1x 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

• 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

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