Unit 6 Trigonometric Identities - CBRH - Corner … Unit 6_student notes .pdf · Unit 6 Trigonometric Identities ... – Replace a “squared” term with a _____ – Write the

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.


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.


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




x

y

x

y

2 2( ) tan 1 secii

cos( ) sec

siniii


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


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, ...


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


Fundamental Trigonometric Identities.

• Reciprocal Quotient Pythagorean

Caution • The Pythagorean identities can be expressed in different ways:


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


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


•In this case we have choices

•What do we do here?

• ____________________________________________________

2secD)

tan

xx

sin cosE)

1 cos sin


• What do we do here?

• _______________________________________

Now we have choice.

1. __________________________________________________

____________________________________________________

tan sinF)

1 cos

x xx


2. __________________________________________________

____________________________________________________

Page 296

# 1 a) d) 3b) c)

4, 7, 8c), 9, 10


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


• _________________________________

• _________________________________________

– 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


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


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


• Which side is the more complicated side? • Lets work on ______________________________________


3 2C) cos cos cos sin

cos 1 sinD) 2sec

1 sin cos



sin 1 cosE)

1 cos sin

2 2 2F) cos sin 2cos 1


sin cosG) sin cos

tan cot

t tt t

t t

sinH) cot csc

1 cos

tt t

t


21 1I) 2 cot

1 cos 1 cost

t t

2

2

1 sec sec 1J)

5sec 22 3sec 5sec


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


PROOF:


• PROOF:

• Replace b by –b in

• PROOF:

• Replace a by in

• PROOF:

• Replace b by –b in

2a


Addition Formula for Tan

PROOF:

Subtraction Formula for Tan


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


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


Identities

3. A) Prove:

B) Prove:

sin cos cos6 3

x x x

cos cos 2cos cos



Double Angle Formulae sin2

cos2

tan2

Examples

1. Find the exact values of:

A) 2sin15ocos15o

2 2) cos sin8 8

B


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


tan2 tan) sin2

tan2 tan

B BB B

B B

) sin sinC

) 2sin sin 23 3 3

D cos


4 2) cos4 8cos 8cos 1E x x x

sin2)Show that can be simplfied tocot

1 cos2

xF x

x


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



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


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


4. Solve: cos 2x + sin2x = 0.7311, for the domain 0° ≤ x < 360°.

Identifying and Repairing Errors

1. Identify and repair the mistake

_____________________________________________________

_____________________________________________________


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


Documents

Unit 6 Trigonometric Identities - CBRH - Corner … Unit 6_student notes .pdf · Unit 6 Trigonometric Identities ... – Replace a “squared” term with a _____ – Write the