Basic Trigonometric Identities

1
Basic Trigonometric Identities
2
Example
Verify the identity sec x cot x csc x.
Solution The left side of the equation contains
the more complicated expression. Thus, we work
with the left side. Let us express this side of
the identity in terms of sines and cosines.
Perhaps this strategy will enable us to transform
the left side into csc x, the expression on the
right.
Apply a reciprocal identity sec x 1/cos x and
a quotient identity cot x cos x/sin x.
Divide both the numerator and the denominator by
cos x, the common factor.
3
Example
Verify the identity cosx - cosxsin2x cos3x..
We worked with the left and arrived at the right
side. Thus, the identity is verified.
4
Guidelines for Verifying Trigonometric Identities

1. Work with each side of the equation independently
   of the other side. Start with the more
   complicated side and transform it in a
   step-by-step fashion until it looks exactly like
   the other side.
2. Analyze the identity and look for opportunities
   to apply the fundamental identities. Rewriting
   the more complicated side of the equation in
   terms of sines and cosines is often helpful.
3. If sums or differences of fractions appear on one
   side, use the least common denominator and
   combine the fractions.
4. Don't be afraid to stop and start over again if
   you are not getting anywhere. Creative puzzle
   solvers know that strategies leading to dead ends
   often provide good problem-solving ideas.

5
Example

• Verify the identity csc(x) / cot (x) sec (x)

Solution
6
Example

• Verify the identity

Solution
7
Example

• Verify the following identity

Solution
8
7.2 Trigonometric Equations
9
Equations Involving a Single Trigonometric
Function
To solve an equation containing a single
trigonometric function Isolate the
function on one side of the equation.
Solve for the variable.
10
Trigonometric Equations
y
y
cos
x
1
y
0.5
x
4
?
2
?
2
?
4
?
1
cos x 0.5 has infinitely many solutions for ?
lt x lt ?
y
y
cos
x
1
0.5
2
?
cos x 0.5 has two solutions for 0 lt x lt 2?
1
11
Example
Solve the equation 3 sin x - 2 5 sin x - 1, 0
x lt 360
Solution The equation contains a single
trigonometric function, sin x.
Step 1 Isolate the function on one side of the
equation. We can solve for sin x by collecting
all terms with sin x on the left side, and all
the constant terms on the right side.
x 210 or x 330
12
Example
Solve the equation 2 cos2 x cos x - 1 0,
0 x lt 2p.
cos x 1/2
x p????? x 2p??p?????p????? x p???
The solutions in the interval 0, 2p) are p/3, p,
and 5p/3.
13
Example

• Solve the following equation for ? is any real
  number.

Solution
14
Example

• Solve the equation on the interval 0,2?)

Solution
15
Example

• Solve the equation on the interval 0,2?)

Solution
16
7.3 Sum and Difference Formulas
17
The Cosine of the Difference of Two Angles

• The cosine of the difference of two angles equals
  the cosine of the first angle times the cosine of
  the second angle plus the sine of the first angle
  times the sine of the second angle.

18
Example

• Find the exact value of cos 15

Substitute exact values from memory or use
special triangles.
Multiply.
Add.
19
Example
Find the exact value of cos 80 cos 20 sin
80 sin 20.
Solution The given expression is the right side
of the formula for cos(? - ?) with ? 80 and ?
20.
cos 80 cos 20 sin 80 sin 20 cos (80 -
20) cos 60 1/2
20
Example

• Find the exact value of cos(180º-30º)

Solution
21
Example

• Verify the following identity

Solution
22
Sum and Difference Formulas for Cosines and Sines
23
Example

• Find the exact value of sin(30º45º)

Solution
24
Sum and Difference Formulas for Tangents

• The tangent of the sum of two angles equals the
  tangent of the first angle plus the tangent of
  the second angle divided by 1 minus their
  product.
• The tangent of the difference of two angles
  equals the tangent of the first angle minus the
  tangent of the second angle divided by 1 plus
  their product.

25
Example

• Find the exact value of tan(105º)

Solution

• tan(105º)tan(60º45º)

26
Example

• Write the following expression as the sine,
  cosine, or tangent of an angle. Then find the
  exact value of the expression.

Solution
27
Review Quiz Answers

•  

28
7.4 Double-Angle and Half-Angle Formulas
29
Double-Angle Identities
30
Three Forms of the Double-Angle Formula for cos2?
31
Power-Reducing Formulas
32
Example

• If sin a 4/5 and a is an acute angle, find the
  exact values of sin 2 a and cos 2 a.

Solution
If we regard a as an acute angle of a right
triangle, we obtain cos a 3/5. We next
substitute in double-angle formulas Sin 2 a 2
sin a cos a 2 (4/5)(3/5) 24/25. Cos 2 a
cos2 a sin2 a (3/5)2 (4/5)2 9/25 16/25
-7/25.
33
Half-Angle Identities
34
Example
Find the exact value of cos 112.5.
Solution Because 112.5 225/2, we use the
half-angle formula for cos ?/2 with ? 225.
What sign should we use when we apply the
formula? Because 112.5 lies in quadrant II,
where only the sine and cosecant are positive,
cos 112.5 lt 0. Thus, we use the - sign in the
half-angle formula.
35
Half-Angle Formulas for
36
Example

• Verify the following identity

Solution
37
7.5 Product-to-Sum and Sum-to-Product Formulas
38
Product-to-Sum Formulas
39
Example

• Express the following product as a sum or
  difference

Solution
40
Text Example
Express each of the following products as a sum
or difference. a. sin 8x sin 3x b. sin 4x
cos x
Solution The product-to-sum formula that we are
using is shown in each of the voice balloons.
a. sin 8x sin 3x 1/2cos (8x - 3x) -
cos(8x 3x) 1/2(cos 5x - cos 11x)
b. sin 4x cos x 1/2sin (4x x) sin(4x -
x) 1/2(sin 5x sin 3x)
41
Sum-to-Product Formulas
42
Example

• Express the difference as a product

Solution
43
Example

• Express the sum as a product

Solution
44
Example

• Verify the following identity

Solution
45
7.6 Inverse Trig Functions
Objective In this section, we will look at the
definitions and properties of the inverse
trigonometric functions. We will recall that to
define an inverse function, it is essential that
the function be one-to-one.
46
7.6 Inverse Trigonometric Functions and Trig
Equations
Domain 1, 1 Range
Domain Range
Domain 1, 1 Range 0, p
47
Let us begin with a simple question
What is the first pair of inverse functions that
pop into YOUR mind?
This may not be your pair but this is a famous
pair. But something is not quite right with this
pair. Do you know what is wrong?
Congratulations if you guessed that the top
function does not really have an inverse because
it is not 1-1 and therefore, the graph will not
pass the horizontal line test.
48
Consider the graph of
Note the two points on the graph and also on the
line y4.
f(2) 4 and f(-2) 4 so what is an inverse
function supposed to do with 4?
By definition, a function cannot generate two
different outputs for the same input, so the sad
truth is that this function, as is, does not have
an inverse.
49
So how is it that we arrange for this function to
have an inverse?
We consider only one half of the graph x gt
0. The graph now passes the horizontal line test
and we do have an inverse
Note how each graph reflects across the line y
x onto its inverse.
50
A similar restriction on the domain is necessary
to create an inverse function for each trig
function.
Consider the sine function.
You can see right away that the sine function
does not pass the horizontal line test.
But we can come up with a valid inverse function
if we restrict the domain as we did with the
previous function.
How would YOU restrict the domain?
51
Take a look at the piece of the graph in the red
frame.
We are going to build the inverse function from
this section of the sine curve because
This section picks up all the outputs of the sine
from 1 to 1.
This section includes the origin. Quadrant I
angles generate the positive ratios and negative
angles in Quadrant IV generate the negative
ratios.
Lets zoom in and look at some key points in this
section.
52
I have plotted the special angles on the curve
and the table.
53
The new table generates the graph of the inverse.
To get a good look at the graph of the inverse
function, we will turn the tables on the sine
function.
The range of the chosen section of the sine is
-1 ,1 so the domain of the arcsin is -1, 1.
54
Note how each point on the original graph gets
reflected onto the graph of the inverse.
etc.
You will see the inverse listed as both
55
In the tradition of inverse functions then we
have
Unless you are instructed to use degrees, you
should assume that inverse trig functions will
generate outputs of real numbers (in radians).
The thing to remember is that for the trig
function the input is the angle and the output is
the ratio, but for the inverse trig function the
input is the ratio and the output is the angle.
56
The other inverse trig functions are generated by
using similar restrictions on the domain of the
trig function. Consider the cosine function
What do you think would be a good domain
restriction for the cosine?
Congratulations if you realized that the
restriction we used on the sine is not going to
work on the cosine.
57
The chosen section for the cosine is in the red
frame. This section includes all outputs from 1
to 1 and all inputs in the first and second
quadrants.
58
The other trig functions require similar
restrictions on their domains in order to
generate an inverse.
yarctan(x)
ytan(x)
59
The table below will summarize the parameters we
have so far. Remember, the angle is the input for
a trig function and the ratio is the output. For
the inverse trig functions the ratio is the input
and the angle is the output.
arcsin(x) arccos(x) arctan(x)
Domain
Range
When xlt0, yarcsin(x) will be in which quadrant?
ylt0 in IV
When xlt0, yarccos(x) will be in which quadrant?
ygt0 in II
ylt0 in IV
When xlt0, yarctan(x) will be in which quadrant?
60
The graphs give you the big picture concerning
the behavior of the inverse trig functions.
Calculators are helpful with calculations (later
for that). But special triangles can be very
helpful with respect to the basics.
Use the special triangles above to answer the
following. Try to figure it out yourself before
you click.
61
OK, lets try a few more. Try them before you peek.
62
Negative inputs for the arccos can be a little
tricky.
From the triangle you can see that arccos(1/2)
60 degrees. But negative inputs for the arccos
generate angles in Quadrant II so we have to use
60 degrees as a reference angle in the second
quadrant.
63
You should be able to do inverse trig
calculations without a calculator when special
angles from the special triangles are involved.
You should also be able to do inverse trig
calculations without a calculator for quadrantal
angles.
Its not that bad. Quadrantal angles are the
angles between the quadrantsangles like
To solve arccos(-1) for example, you could draw a
quick sketch of the cosine section
64
But a lot of people feel comfortable using the
following sketch and the definitions of the trig
ratios.
65
Finally, we encounter the composition of trig
functions with inverse trig functions. The
following are pretty straightforward
compositions. Try them yourself before you click
to the answer.
Did you suspect from the beginning that this was
the answer because that is the way inverse
functions are SUPPOSED to behave? If so, good
instincts but.
66
Consider a slightly different setup
This is also the composition of two inverse
functions but
Did you suspect the answer was going to be 120
degrees? This problem behaved differently because
the first angle, 120 degrees, was outside the
range of the arcsin. So use some caution when
evaluating the composition of inverse trig
functions.
The remainder of this presentation consists of
practice problems, their answers and a few
complete solutions.
67
Find the exact value of each expression without
using a calculator. When your answer is an angle,
express it in radians. Work out the answers
yourself before you click.
68
Answers for problems 1 9.
Negative ratios for arccos generate angles in
Quadrant II.
69
(No Transcript)
70
Review Answers for Test

•