Trigonometry 3 презентация

Preview activity: Inverse of trig functions

Preview activity: Inverse of trig functions solution.

Trigonometry 3
Graphs of the inverse of trig fun
Solve basic equations
Inverse of Trig functions

Introduction
There are several applications for the inverse of the trig functions…
Finding the angle of elevation.

Introduction

Foundation Year Program
3.3.1 Inverse functions arcsin(x), arccos(x) and arctan(x) .
The inverse function performs the opposite operation to the function. It takes elements of the range and maps them back into elements of the domain. Inverse functions only exist for one-to-one functions.

Foundation Year Program
Trigonometric functions
sin(x), cos(x), tan(x)
have inverse functions if their domains are restricted so that they are one-to-one functions!

Foundation Year Program
The domain and the range

Foundation Year Program
Arcsin(x) is an angle α, in the interval
for which sin(α) =x
arcsin(x) function
Reflect in the line y=x

Foundation Year Program
Arccos(x) is an angle α, in the interval
for which cos(α)=x
arccos(x) function
Reflect in the line y=x

Foundation Year Program
Arctan(x) is an angle α, in the interval
for which tan(α)=x
arctan(x) function
Reflect in the line y=x

Foundation Year Program
Example 1:
We need to solve
on the interval
Answer:

Foundation Year Program
Example 2:
We need to solve
on the interval
Answer:

Foundation Year Program
Your turn!

Foundation Year Program
We need to solve
on the interval
Answer:
Your turn!

Example 3:
A 40-ft ladder lends against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle formed by the ladder and the building?

Example 3:

A function and its inverse
Foundation Year Program

A function and its inverse
Foundation Year Program

3.3.3 Solving
Trigonometric Equations
Use standard algebraic techniques.
Look for factoring and collecting like terms.
Isolate the trig function in the equation.
Use the inverse trig functions to assist in determining
solutions.
To solve trigonometric equations:

Solving
Trigonometric Equations
For all problems,(in this presentation)
The solution interval
Will be
[0, 2π)
You are responsible for checking your solutions
back into the original problem!

Example 5:
: Solve:
Isolate cos x using algebraic skills.
QI
QIV
Note: cosine is positive in
Quad I and Quad IV.
Note: The reference angle is π/3.

Example 6:
Solve:
Note: Since there is a ± ,
all four quadrants
hold a solution with π/4
being the reference
angle.
Q1
QII
QIII
QIV

Example 7:
Solve:

Example 7:
No solutions

Additional exercises
Try these:
1.
2.
3.
Solution

Example 8:
Solve:
Factor the quadratic equation.
Set each factor equal to zero.

Example 8:
Determine the correct quadrants
for the solution(s).

Example 9:
Solve:
Square both sides of the equation
in order to change sine into terms
of cosine giving only one trig
function to work with.
Replace sin2x with 1 – cos2x
Simplify, factor, then solve for cos x

Example 9:
Is 3π/2 a solution?
No, its not, because it does not
check in the original equation.

Solving
Trigonometric Equations
What you should know:
How to use algebraic techniques to solve
trigonometric equations.
How to solve quadratic trigonometric equations
by factoring or the quadratic formula.
How to solve trigonometric equations involving
multiple angles.

Proving some useful identities

Proving some useful identities

Learning outcomes
3.3.1 Compute the values of inverse trigonometric functions.
3.3.2 Sketch the graphs of inverse trigonometric functions.
3.3.3 Solve basic trigonometric equations.