5
Trigonometric
Identities
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5.1-1
Prayer
Let us
remember that
we are in the
Most Holy
Presence of
God.
In the name of
the Father, and
of the Son and
of the Holy
Spirit, amen.
St John Baptiste de La Salle, pray
for us. Leave Jesus in our hearts,
forever.
SCHEDULE
DATE
October 12
October 13
October 14
October 15
October 19
ACTIVITY
TRIGONOMETRIC IDENTITIES
October 20
October 21
October 22
ONLINE QUIZ (TRIGONOMETRIC IDENTITIES )730-930A
RECOLLECTION
BREATHER
October 26
October 27
October 28-30
November 2
November 3
November 4
TRIGONOMETRIC EQUATIONS & OBLIQUE TRIANGLES
MODULE 6 PROBLEM SET
LA SALLIAN DAYS
HOLIDAY
MODULE 6 PROBLEM SET (DISCUSSION)
ONLINE QUIZ (MODULE 6 )1030-1230P
November 9-12
MIDTERM EXAMS
MODULE 5 PROBLEM SET
MODULE 5 PROBLEM SET (DISCUSSION)
5.1-3
5 Targeted Outcomes
1.To demonstrate that the trigonometric functions of angles
are related to each other through trigonometric identities.
2. Demonstrate the evaluation and graph sketching of
inverse trigonometric functions.
Learning Objectives
1. Familiarize with the different trigonometric identities.
2. Perform the evaluation of inverse trigonometric
identities.
3. Sketching the graph of inverse trigonometric functions.
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5.1-4
5 Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine
and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
5.7 Graphs of Trigonometric Functions
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5.1-5
“All Students Take Calculus”
Note
In trigonometric identities, θ can be
an angle in degrees, an angle in
radians, a real number, or a variable.
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1.1-6
5.1-6
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT
Example 1
and θ is in quadrant II, find each function
If
value.
c
5
q
-3
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1.1-7
5.1-7
Fundamental Identities
Reciprocal Identities
Quotient Identities
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1.1-8
5.1-8
Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
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1.1-9
5.1-9
Proving Trigonometric Identities
Suggestions...
1. Learn well the formulas given above (or at least, know how to find them quickly). The
better you know the basic identities, the easier it will be to recognise what is going on in
the problems.
2. Work on the most complex side and simplify it so that it has the same form as the simplest
side.
3. Don't assume the identity to prove the identity. This means don't work on both sides of
the equality sign and try to meet in the middle. Start on one side and make it look like
the other side.
4. Many of these come out quite easily if you express everything on the most complex side
in terms of sine and cosine only.
5. In most examples where you see power 2 (that is, 2), it will involve using the identity sin2
θ + cos2 θ = 1 (or one of the other 2 formulas that we derived above).
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1.1-10
5.1-10
Example 2
PROVING IDENTITIES
Prove the following trigonometric equations are identities:
Note: LHS means “Left Hand Side” and RHS means “Right Hand Side” of
the equation.
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1.1-11
5.1-11
Example 3
PROVING IDENTITIES
Prove the following trigonometric equations are identities:
Note: LHS means “Left Hand Side” and RHS means “Right Hand Side” of
the equation.
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1.1-12
5.1-12
Caution
When working with trigonometric
expressions and identities, be sure
to write the argument of the function.
For example, we would not write
An argument such as θ
is necessary.
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1.1-13
5.1-13
Double-Angle Identities
We can use the cosine sum identity to derive
double-angle identities for cosine.
Cosine sum identity
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5.5-14
Double-Angle Identities
There are two alternate forms of this identity.
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5.5-15
Double-Angle Identities
We can use the sine sum identity to derive a
double-angle identity for sine.
Sine sum identity
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5.5-16
Double-Angle Identities
We can use the tangent sum identity to derive a
double-angle identity for tangent.
Tangent sum identity
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5.5-17
Double-Angle Identities
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1.1-18
5.5-18
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ
Example 4
and sin θ < 0, find sin 2θ, cos 2θ, and
Given
tan 2θ.
The identity for sin 2θ requires sin θ.
3
q
5
-4
-
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Any of the three
forms may
be used.
1.1-19
5.5-19
Example 4
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ (cont.)
Alternatively, find tan 2θ by finding the quotient of
sin 2θ and cos 2θ.
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1.1-20
5.5-20
Example 5
VERIFYING A DOUBLE-ANGLE IDENTITY
Verify that
is an identity.
Quotient identity
Double-angle
identity
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1.1-21
5.5-21
Product-to-Sum Identities
The identities for cos(A + B) and cos(A – B) can be
added to derive a product-to-sum identity for
cosines.
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5.5-22
Product-to-Sum Identities
Similarly, subtracting cos(A + B) from cos(A – B)
gives a product-to-sum identity for sines.
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5.5-23
Product-to-Sum Identities
Using the identities for sin(A + B) and sine(A – B)
gives the following product-to-sum identities.
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5.5-24
Product-to-Sum Identities
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1.1-25
5.5-25
Example 6
USING A PRODUCT-TO-SUM IDENTITY
Write 4 cos 75° sin 25° as the sum or difference of
two functions.
4{ ½ [sin(750 + 250)-sin(750 – 250)]}
Click to add text
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1.1-26
5.5-26
Sum-to-Product Identities
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1.1-27
5.5-27
Half-Angle Identities
We can use the cosine sum identities to derive halfangle identities.
Choose the appropriate sign depending on the
quadrant of
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5.6-28
Half-Angle Identities
Choose the appropriate sign depending on the
quadrant of
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5.6-29
Half-Angle Identities
There are three alternative forms for
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5.6-30
Half-Angle Identities
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5.6-31
Double-Angle Identities
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1.1-32
5.6-32
Example 7
USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of cos 15° using the half-angle
identity for cosine.
Choose the positive
square root.
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1.1-33
5.6-33
Example 8
VERIFYING AN IDENTITY
Verify that
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is an identity.
1.1-34
5.6-34
5.7 Graphs of Trigonometric
Functions
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1.1-35
5.1-35
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3
2
-
-

1

2
2

3
2
2
5
2
x
-1
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36
Arcsine Function
Arcsine function is an inverse of the sine function denoted by
sin-1x. It is represented in the graph as shown below:
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37
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
cos x
1
2
0
-1
0
1
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = cos x
y
3
2
-
-

1

2
2

3
2
2
5
2
x
-1
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38
Arccosine Function
Arccosine function is the inverse of the cosine function
denoted by cos-1x. It is represented in the graph as shown
below:
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39
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x =
.
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x

x  k  + (k   )
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x = k  + (k   )
2

2
- 3
2
3
2
x
-
2
period: 
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40
Arctangent Function
Arctangent function is the inverse of the tangent function
denoted by tan-1x. It is represented in the graph as shown
below:
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41
Graph of the Cotangent Function
cos x
To graph y = cot x, use the identity cot x =
.
sin x
At values of x for which sin x = 0, the cotangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = cot x
y = cot x
1. domain : all real x
x  k  (k   )
2. range: (–, +)
3. period: 
4. vertical asymptotes:
x = k  (k   )
vertical asymptotes
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-
3
2
- 

 3
2
2
2
-
x = -
x=0
x=
x
2
x = 2
42
Arccotangent (Arccot) Function
Arccotangent function is the inverse of the cotangent function
denoted by cot-1x. It is represented in the graph as shown
below:
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43
Graph of the Secant Function
The graph y = sec x, use the identity sec x =
1
.
cos x
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y = sec x
Properties of y = sec x
1. domain : all real x

x  k + ( k   )
2
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:

x = k  + (k   )
2
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4
y = cos x
x
-


2
2

3
2
2
5 3
2
-4
44
Arcsecant Function
Arcsecant function is the inverse of the secant function
denoted by sec-1x. It is represented in the graph as shown
below:
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45
Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x =
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
4
y = csc x
1. domain : all real x
x  k  (k   )
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
x = k  (k   )
where sine is zero.
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x
-


2
2

3 2
2
5
2
y = sin x
-4
46
Arccosecant Function
What is arccosecant (arccsc x) function? Arccosecant function
is the inverse of the cosecant function denoted by cosec -1x. It
is represented in the graph as shown below:
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47
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5.5-48