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1 | P a g e Math 3200 Unit 6
Unit 6
Trigonometric Identities • Prove trigonometric identities
• Solve trigonometric equations
Prove trigonometric identities, using:
• Reciprocal identities
• Quotient identities
• Pythagorean identities
• Sum or difference identities
– (restricted to sine, cosine and tangent)
• Double-angle identities
– (restricted to sine, cosine and tangent)
TRIG IDENTITIES
• You should be able to explain the difference between a
trigonometric __________ and a trigonometric _____________.
• An __________ is true for ____________________, whereas
an ____________ is only true for a _______________ of the
permissible values.
• This difference can be demonstrated with the aid of graphing
technology.
2 | P a g e Math 3200 Unit 6
For example: 1
sin ,0 3602
x x
• This can be solved by using the graphs of ___________________
• The solutions to 1
sin ,0 3602
x x
are x = ___ and x = ____, which are the
______________________________
______________________________.
• Thus this is _____________________
because it is ____________________
_____________________________.
Solve: sin x = tan x cos x. • This can be solved by using the graphs of:
y = sin x and y = tan x cos x
• These graphs are _________ identical.
– The only differences in the graphs occur at the points _____
______________.
– Why? _________________________________________
• Therefore, sin x = tan x cos x is an ________ since the
expressions are ______________ for all permissible values.
3 | P a g e Math 3200 Unit 6
x
y
• Note: There may be some points for which identities are
_______________________.
• These ____________________ values for identities occur
where one of the expressions is ______________
• In the previous example, y = tan x cos x is not defined when
________________________ since _________ is undefined at
these values.
• Non-permissible values often occur when a trigonometric
expression contains:
– A ______________________, resulting in values that give a
denominator of ________
– ___________________________________, since these
expressions all have non-permissible values in their domains.
Practice: Determine graphically if the following are identities. Use Technology
Identify the non-permissible values.
• Non-permissible values? Is this an identity?
( ) sin cos tan 2sini
4 | P a g e Math 3200 Unit 6
• Non-permissible values? Is this an identity?
• Non-permissible values? Is this an identity?
x
y
x
y
2 2( ) tan 1 secii
cos( ) sec
siniii
5 | P a g e Math 3200 Unit 6
We can also verify numerically that an identity is valid by substituting
numerical values into both sides of the equation.
•Example: Verify whether the following are identities.
A) B)
(use degrees) (use radians)
C) D)
(use degrees) (use radians)
NOTE:
•This approach is ______________ to conclude that the equation is
an identity because only a _____________ of values were substituted
for θ, and there may be a certain group of numbers for which this
identity ________________.
•To prove the identity is true using this method would require verifying
_____ of the values in the domain (__________________).
•This type of reasoning is called __________________.
sin cos 2 2sin cos 1
2 2tan 1 sec 2 2cot 1 csc
6 | P a g e Math 3200 Unit 6
Proofs! • A proof is a __________________ that is used to show the
validity of a mathematical statement.
• Deductive reasoning occurs when general ___________________
are __________ to specific situations.
• Deductive reasoning is the process of _____________________
based on facts that have already been shown to be ______.
• The facts that can be used to prove your conclusion deductively
may come from accepted ______________________________.
• The truth of the premises ________________ the truth of the
conclusion.
Find the fifth term in the sequence • Inductive Reasoning
1. 3, 5, 7, 9, . . .
2. 3, 12, 27, 48, ...
3. 7, 14, 21, 28, ...
7 | P a g e Math 3200 Unit 6
Deductive Reasoning
1. tn =2n + 1
2. tn = 3n2
3. Dates of Wednesdays in 2015 year
What is the next number in this sequence?
• 15, 16, 18, 19, 25, 26, 28, 29, ______
Trig Proofs • Trig proofs (and simplifications of trig expressions) are based on
the definition of the 6 trigonometric functions and the
Fundamental Trigonometric Identities.
Definition of the 6 trigonometric functions
• Sine fn:
• Cosine fn
• Tangent fn
• Cotangent
• Secant fn
• Cosecant fn
8 | P a g e Math 3200 Unit 6
Fundamental Trigonometric Identities.
• Reciprocal Quotient Pythagorean
Caution • The Pythagorean identities can be expressed in different ways:
9 | P a g e Math 3200 Unit 6
Simplify expressions using the Pythagorean identities,
the reciprocal identities, and the quotient identities
• Strategies that you might use to begin the simplifications:
– Replace a “squared” term with a ______________________
– Write the expression in terms of _____________________
– For expressions involving addition or subtraction, it may be
necessary to use a _____________________ to simplify a
fraction
– _____________
– Multiply by a __________ to obtain a _________________
–
• You may also be asked to determine any __________________
values of the variable in an expression.
For example, identify the non-permissible values of θ in
and then simplify the expression.
Solution:
______________________________________________
__________________________________
sin cos cot
Simplify : sin cos cot
10 | P a g e Math 3200 Unit 6
NOTE: • Students often find simplifying trigonometric expressions more
challenging than proving trigonometric identities because they may
be uncertain of when an expression is simplified as much as
possible.
• However, developing a good foundation with simplifying expressions
makes the transition to proving trigonometric identities easier.
• Simplify the following
• In this case we use a ____________________________________
•In this case we ________________________________________
A) sin secx x
21 cosB)
sin
2C) sec sec sin
11 | P a g e Math 3200 Unit 6
•In this case we have choices
•What do we do here?
• ____________________________________________________
2secD)
tan
xx
sin cosE)
1 cos sin
12 | P a g e Math 3200 Unit 6
• What do we do here?
• _______________________________________
Now we have choice.
1. __________________________________________________
____________________________________________________
tan sinF)
1 cos
x xx
13 | P a g e Math 3200 Unit 6
2. __________________________________________________
____________________________________________________
Page 296
# 1 a) d) 3b) c)
4, 7, 8c), 9, 10
14 | P a g e Math 3200 Unit 6
Warm UP
• Factor and simplify
Proving Identities
• The fundamental trigonometric identities are used to establish
other relationships among trigonometric functions.
• To ______________________, we show that one side is equal
to the other side.
• Each side is manipulated __________________ of the other
side.
– It is ____________ to perform operations ___________
_________________, such as
• _________________________________________
_________________________________________
2
2
sin sin cos1.
sin
x x x
x
2tan 3tan 4
2.sin tan sin
x xx x x
15 | P a g e Math 3200 Unit 6
• _________________________________
• _________________________________________
– These operations are only possible if the equation is _____.
– Until we verify, or prove the identity to be ______, we do
not know if both sides are ________.
Prove that the following are Identities using the definitions of
the trig function on the unit circle
1) cos
secA
sinB) tan
cos
2 2C) sin cos 1 2 2D) tan 1 sec
16 | P a g e Math 3200 Unit 6
Guidelines for Proving Trigonometric Identities
• We usually start with side that contains the more ___________
expression.
• If you substitute one or more __________________________
on the more complicated side you will often be able to rewrite it
in a form identical to that of the other side.
• Rewriting the complicated side in terms of _________________
is often helpful.
• If sums or differences appear on one side, use ______________
______________ and combine fractions
• In other cases ______________ is useful.
• It may be necessary to multiply a fraction by a _____________
to obtain a ________________________
• There is ___________________ that can be used to prove
every identity.
• In fact there are often ______ different methods that may be
used.
• However, one method may be __________ and _____________
than another.
• The more identities _____ prove, the more confident and
efficient you will become.
• ___________________________________
• DON’T BE AFRAID to _____________________ over again if
you are not getting anywhere.
• Creative puzzle solvers know that strategies leading to dead ends
often provide good problem-solving ideas
17 | P a g e Math 3200 Unit 6
Prove the following
• Which side is the more complicated side?
• Lets work on ______________________________________
• Which side is the more complicated side?
• Lets work on ______________________________________
A) sec cot cscx x x
B) sin tan cos secx x x x
18 | P a g e Math 3200 Unit 6
• Which side is the more complicated side? • Lets work on ______________________________________
• Which side is the more complicated side? • Lets work on ______________________________________
3 2C) cos cos cos sin
cos 1 sinD) 2sec
1 sin cos
19 | P a g e Math 3200 Unit 6
• Which side is the more complicated side? • Lets work on ______________________________________
sin 1 cosE)
1 cos sin
2 2 2F) cos sin 2cos 1
20 | P a g e Math 3200 Unit 6
sin cosG) sin cos
tan cot
t tt t
t t
sinH) cot csc
1 cos
tt t
t
21 | P a g e Math 3200 Unit 6
21 1I) 2 cot
1 cos 1 cost
t t
2
2
1 sec sec 1J)
5sec 22 3sec 5sec
22 | P a g e Math 3200 Unit 6
OTHER TRIG STUFF
• Even-Odd Identities (Negative Angle):
• Addition and Subtraction Rules:
PROOF:
• This one of those “interesting proofs”.
• We need to use the:
• Law of Cosines
•And the distance formula between 2 points
23 | P a g e Math 3200 Unit 6
PROOF:
24 | P a g e Math 3200 Unit 6
• PROOF:
• Replace b by –b in
• PROOF:
• Replace a by in
• PROOF:
• Replace b by –b in
2a
25 | P a g e Math 3200 Unit 6
Addition Formula for Tan
PROOF:
Subtraction Formula for Tan
26 | P a g e Math 3200 Unit 6
Applications of the Angle Addition Formulae • Finding exact values
• Deriving double and half angle formula
• Proving Identities
• In Calculus:
• Trig derivatives
• Trig substitution in integration.
• Find the exact values of:
A) cos 15o B) sin 75o
C) D)
E) sin 60o cos 30o + sin 30o cos 60o F)
How can we verify that this is true?
sin12
7tan
12
tan15 tan30
1 tan15 tan30
o o
o o
27 | P a g e Math 3200 Unit 6
G) A and B are both in Quadrant II, 513
cosA and 35
sinB .
Determine the exact value of cos A B .
2. Simplify
A) sin sin2 2
B) tan
28 | P a g e Math 3200 Unit 6
Identities
3. A) Prove:
B) Prove:
sin cos cos6 3
x x x
cos cos 2cos cos
29 | P a g e Math 3200 Unit 6
30 | P a g e Math 3200 Unit 6
Double Angle Formulae sin2
cos2
tan2
Examples
1. Find the exact values of:
A) 2sin15ocos15o
2 2) cos sin8 8
B
31 | P a g e Math 3200 Unit 6
2. Simplify:
3. PROVE:
2
4tan) sin cos )
2 2 1 tan
x xA B
21 cos2) tan
1 cos2
AA A
A
2cos2 sin)
sin2
x xC
x
32 | P a g e Math 3200 Unit 6
tan2 tan) sin2
tan2 tan
B BB B
B B
) sin sinC
) 2sin sin 23 3 3
D cos
33 | P a g e Math 3200 Unit 6
4 2) cos4 8cos 8cos 1E x x x
sin2)Show that can be simplfied tocot
1 cos2
xF x
x
34 | P a g e Math 3200 Unit 6
4. Suppose:
Find the exact value of:
A) sin 2 B) cos 2 C) tan2
Half Angle Formulae Not on Public but good to know
Consider:
cos2 = 1 – 2sin2 cos2 = 2cos2- 1
Let Let
• Examples:
Find the exact value of:
A) sin 15o B) cos 75o
1 3sin
4 2 2and
35 | P a g e Math 3200 Unit 6
36 | P a g e Math 3200 Unit 6
Last Section for Chapter 6 (6.4)
Solve, algebraically and graphically, first and second degree
trigonometric equations
• The identities encountered earlier in this unit can now be applied
to solve trigonometric equations.
Examples:
1. Find the solutions of . for 0° ≤ x < 360°.
Solution: Graphically
A) Identify each curve B) What are the
points of intersection?
Solution: Algebraically
What are the solutions with an unrestricted domain, in radians?
sin2 3cosx x
37 | P a g e Math 3200 Unit 6
2. Solve for 0° ≤ x ≤ 360°, giving exact solutions
where possible.
• Write the general solution in degrees and radian measure.
3. Solve the trigonometric equation shown below for :
cos2 1 cosx x
0 2x
32
sin3 cos cos3 sinx x x x
38 | P a g e Math 3200 Unit 6
4. Solve: cos 2x + sin2x = 0.7311, for the domain 0° ≤ x < 360°.
Identifying and Repairing Errors
1. Identify and repair the mistake
_____________________________________________________
_____________________________________________________
39 | P a g e Math 3200 Unit 6
2. A student’s solution for tan2 x = sec x tan2 x for 0 ≤ x < π is shown
below:
• Identify and explain the error(s).
• How many mark should the student get if this question was worth
4 marks?
• Provide the correct solution
40 | P a g e Math 3200 Unit 6