Download A. Logarithmic Identities and Proving Log Identities…

Pre-Calculus 12
Unit: Identities
Chapter 1 Notes
A. Logarithmic Identities and Proving Log Identities…
Equations that are true for all values of the variable are called identities. Equations with logs are log identities. We can
manipulate equations using the “Log Laws” we are now used to work with and create many different log identities
(there are as many different identities as there are equations!). Some log identities (or log equations) are handy to have
“in the back of our mind,” we can use them to solve log equations that have a finite set of solutions (remember: log
identities are true for all values of the variable).
To prove an identity (which is just an equation), is to show the mathematical steps involved in
making one side the same as the other. Easier said than done! You will need to have all the “algebra
tricks” (factoring, multiplying top and bottom, etc.) from your math background and your new “log
tricks” you have learned in this unit.
1
Ex 1: Given the Identity:
log a ( )  log a x
x
a) Verify that it is true numerically.
b) Determine the values of x for which each side is defined.
c) Verify the identity graphically (when a = 10).
d) Prove the identityfor any base a and any positive value of x.
Ex 2: Prove the following identities and state the values of x for which each side is defined:
1
log a (b c )  log a c
a)
loga(x + y) + loga(x – y) = loga(x2 – y2)
b)
b

c)
d)

1
1
1


log a x log b x log ab x
Pre-Calculus 12
Unit: Identities
Ch.1 Notes - Page 2
B. Trigonometric Identities and Proving Trig Identities…
There are three ways to show that an identity works for the values you want. The first two ways
verify that the identity works for the values you try, however you are never sure that there aren’t
some “rogue” values, which may invalidate your identity. One sure way to know the identity works
for all values is to prove an identity using an Algebraic Proof.
Verify Numerically: Insert values in for x and show it works
Ex 1a
sin x tan x sec x
= tan2x
Verify Graphically: Graph the two sides simultaneously and show the overlap
Ex 1b
sin x tan x sec x
Algebraic Proof:
= tan2x
This is the way that shows the identity is true for all values of x
C. Formal Proofs of Identities
Note: there are some formal and informal rules we follow when doing proofs:
Formal Rules:
 Work with only one side - no back and forth
(you can try both sides to get started or for ideas, but finish only one side)
 Each step under the one before
 No skipping steps (one idea/manipulation per step)
 No cross multiplying ( the is no = sign, so not cross mult )
 Conclude with LS or RS
Informal “Rules”:
 Start with the most complicated side
 Change everything to sin and cos
 Use a common denominator for complex fractions
(turn dividing rational expressions into multiply and cancel top to bottom)
 Look for (and use) obvious substitutions
 Look for (and use) conjugates
Pre-Calculus 12
Ex 1c
Ex 2
Prove:
Prove:

Unit: Identities
sin x tan x sec x
sin x  cos x cot x
cos x csc x
=
=

2
tan x
secx
Ch.1 Notes - Page 3
Pre-Calculus 12
Ex 3
Prove:
1  cos x
sin x

Ex 3
Prove:

Unit: Identities
=
1
csc x  cot x
=
1  sin x
cos x

1
sec x  tan x

Ch.1 Notes - Page 4
Pre-Calculus 12
Unit: Identities
Ch.1 Notes - Page 5
PRACTICE PROBLEMS
1.
2.
and state any restriction on the variable(s):
Pre-Calculus 12
Unit: Identities
Ch.1 Notes - Page 6
3. Prove the following identities and state any restriction on the variable(s):
a)
c)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
b)
d)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
(Bonus type – done by manipulating the def’n)
3.
Pre-Calculus 12
Unit: Identities
Ch.1 Notes - Page 7
Pre-Calculus 12
4.
5.
Unit: Identities
Ch.1 Notes - Page 8
Pre-Calculus 12
6.
7.
8.
Unit: Identities
Ch.1 Notes - Page 9
Pre-Calculus 12
Unit: Identities
Chapter 2 Notes
A. Proofs with Sum and Difference Identities
sin( + ) = sin  cos  + cos  sin 
sin( - ) = sin  cos  - cos  sin 
cos( + ) = cos  cos  - sin  sin 
cos( - ) = cos  cos  + sin  sin 
On the formula sheet you will find four sum and difference identities that work with sine and cosine. We
can use these identities to prove further identities as well, but they have other uses as well. First lets try
a proof using one of these identities.
Ex 1
Prove:
Ex 2
Prove:
sin (A + B) + cos (A – B) = (sin A + cos A)(sin B + cos B)
sin (A + B) =
cot A + cot B
csc A csc B
Pre-Calculus 12
Unit: Identities
Ch.2 Notes - Page 2
B. Using Sum & Difference Identities to find other Exact Values
Up to now we have worked with special angles such as π/6, π/4, π/3, and so on. We have been able to
find exact answers to the trigonometric functions (without using a calculator), with the help of special
triangles. Now, with the help of these new sum & difference identities we can use them to find exact
values of a few other related angles.
What other angles?
Given (A + B) & (A – B) to work with, what other angles can you make from as π/6, π/4, π/3?
Now that we can make other angles out of our familiar angles we can use the sum & difference
identities to find new exact values. The identities work as a substitution, decompose the angles you want
values for into ones we know (if possible) and then use the identities to find the values.
Ex 1
Find the exact value of
sin

12

Ex 2
Find the exact value of
cos
5
12
tan
7
12

Ex 3
Find the exact value of

Pre-Calculus 12
Unit: Identities
Ch.2 Notes - Page 3
C. Using Sum & Difference Identities to simplify expressions:
We can use the identities backwards and in other ways as well…
Ex 4
Find the exact value of:

Ex 5
sin

Find a simplified expression for:




cos - cos sin
12
6
12
6


cos(π – x)
Pre-Calculus 12
Unit: Identities
Ch.2 Notes - Page 4
D. Using Sum & Difference Identities to solve problems:
Ex.1 Given that A is in Quadrant II and sin A = 3/5, and B is in Quadrant III and cot B = 12/5, what is
the value of cos(A - B) ?
Ex.2 Given that A is in Quadrant II, csc A = 25/7, B is in Quadrant IV, and sec B = 17/8; what is the
value of tan(A + B) ?
Pre-Calculus 12
Unit: Identities
Worksheet:
1.
Ch.2 Notes - Page 5
Pre-Calculus 12
Unit: Identities
Ch.2 Notes - Page 6
2.
3.
4.
5. Given that A is in Quadrant II, csc A = 29/20, B is in Quadrant III, and tan B = 12/35; what is the
value of cot(A - B) ?
Pre-Calculus 12
Unit: Identities
Chapter 3 Notes
Double Angle Identities and Proofs
Here are a new set of identities; the Double Angle Identities. You will notice how the angles in the
identities are multiplied by two:
sin 2x = 2sin x cos x
cos 2x = cos2 x - sin2 x
Together with the Pythagorean identity we get:
cos 2x = 2cos2 x – 1
cos 2x = 1 - 2sin2 x
A. Using Double Angle Identities to find other exact trig values
Ex 1:
sin

12
Ex 2:

cos
5
8

B. Using Double Angle Identities to Simplify Expressions
We can use the double angle identities to do some simplifying and rewriting of expressions. When using
double angle identities, they are often used best by straight substitution. Note, there are many questions
that look like they are utilizing double angles, but they need some refining first…
Ex 1: Simplify - in terms of sin x & cos x
a)
sin(4x)
b)

x
x
8sin( )cos( )
2
2

c)
cos(4x)
Pre-Calculus 12
Ex 2: Simplify as much as possible:
a) 6cos2(2x) - 3
Unit: Identities
b)
Ex 3: Solve for x, with an exact value for x, where 0 ≤ x<2π:
1.
sin 2x – 2cos x = 0
2.
Ch.3 Notes - Page 2
(cos3x + sin3x)(cos3x – sin3x)
sin x tan 2x = sin x
Pre-Calculus 12
Unit: Identities
Ch.3 Notes - Page 3
C. Double Angle Identities and Proofs
Here are a new set of identities; the Double Angle Identities. You will notice how the angles in the
identities are multiplied by two:
sin 2x = 2sin x cos x
cos 2x = cos2 x - sin2 x
Together with the Pythagorean identity we get:
cos 2x = 2cos2 x – 1
cos 2x = 1 - 2sin2 x
Let’s see if we can use all the identities to prove various trig identities
cot x  tan x
Ex 1
Prove:
=
(and state any restrictions on x)
cos 2 x
cot x  tan x
Pre-Calculus 12
Ex 2
Prove
cos 2 x
1  sin 2 x
Unit: Identities
=
1  tan x
1  tan x
Ch.3 Notes - Page 4
(and state any restrictions on x)
Pre-Calculus 12
Ex 3
Prove:
sec x
1  sin x
Unit: Identities
=
1  sin x
cos 3 x
Ch.3 Notes - Page 5
(and state any restrictions on x)
Pre-Calculus 12
Unit: Identities
Worksheet:
1.
2.
b.
a.
Ch.3 Notes - Page 6
Pre-Calculus 12
c.
3.
4.
Unit: Identities
d.
Ch.3 Notes - Page 7
Pre-Calculus 12
5.
6.
7.
Unit: Identities
Ch.3 Notes - Page 8
Pre-Calculus 12
Unit: Identities
A. Proving Identities Practice:
Chapter 4 Notes
The “Genesis” Identities – cannot be proven with the LS/RS structured method. Restrictions?
log x
a)
sin2x + cos2x = 1
b)
log a x 
log a
Ex.1: From sin2x + cos2x = 1 we get the following. Prove and state any restrictions:
 b)
a)
1 + tan2x = sec2x
1 + cot2x = csc2x
Ex.2: From sin(a+b) = sinacosb +sinbcosa (& others)we get these. Prove and state any restrictions:
a)
sin2x = 2sinxcosx
b)
cos2x = cos2x - sin2x
Ex.3: Prove and state any restrictions:
a)
sin22x + cos22x = 1
b)
sin2x + cos2x = 1
Pre-Calculus 12
Unit: Identities
Worksheet:
1. Prove and state any restrictions:
Ch.4 Notes - Page 2
Pre-Calculus 12
2. Prove and state any restrictions:
Unit: Identities
Ch.4 Notes - Page 3
Unit IDENTITIES: Review
-1-
Pre-Calculus 12 - Mr. Muller
SHOW EVERY STEP!
Name:___________________________ DATE: ________
A. Simplify the following expressions in terms of sine and/or cosine only:
1.
(sec x csc x – tan x)(cos x - sec x)
2.
sin x – csc x
sec x – cos x
B. Find the exact value of the following Trigonometric Expressions:

7
1.
sin
2.
cos
12
12


(this one is different)
3.
cos
7
8

C. Find the exact value of the following Trigonometric Expressions:
1.
cos 15˚ cos 75˚ – sin 15˚ sin 75˚
2.
sin 72˚ cos 12˚ – cos 72˚ sin 12˚
D. Evaluate Each Expression
1.
Given that tan A = 3/4 is in Quadrant I,
and cos B = –5/13 is in Quadrant II, what
is the value of sin (A – B) ?
2.
Given that csc A = –13/12 is in Quadrant III,
and cos B = 3/5 is in Quadrant IV, what
is the value of tan (A + B) ?
Unit IDENTITIES: Review
-2E. Find the exact value of the following Trigonometric Expressions:
1.
cos222.5˚ – sin222.5˚
2.
5 – 10cos215˚
Pre-Calculus 12 - Mr. Muller
11sin2
3.



– 11cos2
12
12

F. Given the information about the angle find the exact value of the following:
1.
If tan x = 5/12, and x is in Quadrant III,
2.
If tan x = 3/4, what is the value of tan 2x ?
what is the value of sin 2x ?
G. Solve for x, where 0 ≤ x < 2π :
1.
sin 2x = 2tan x
2.
cos x cos 2x – cos2 x = 0
(Hint: write in terms of cosine only, then solve)
Unit IDENTITIES: Review
H. Solve for x where 0 ≤ x < 2π :
1.
2sin2 x – cos x + 4 = 5
(Hint: write in terms of cosine only, then solve)
-32.
Pre-Calculus 12 - Mr. Muller
cos 2x – 1 = sin x
(Hint: write in terms of sine only, then solve)
I. Prove the following Trigonometric Identities and state any restrictions on the variable(s):
1.
2sin2 x = tan x
2.
cot x =
1 + cos 2x
sin 2x
sin 2x
Restrictions:
Restrictions:
Unit IDENTITIES: Review
-4Pre-Calculus 12 - Mr. Muller
J. Prove the following Trigonometric Identities and state any restrictions on the variable(s):
1.
cos (x + y ) = cot y – tan x
2.
csc x
= cos x
cos x sin y
cot x + tan x
Restrictions:
Restrictions:
Unit IDENTITIES: Review
-5Pre-Calculus 12 - Mr. Muller
K. Prove the following Trigonometric Identities and state any restrictions on the variable(s):
1.
2 cos3 x – cos x = cos2 x – sin2 x
2.
1 – sin x
=
1
sec x
cos x
sec x + tan x
Restrictions:
Restrictions:
L. Prove the following Logarithmic identities and state any restrictions on the variable(s):
1.
log x + log(x – 1) = log(x2 – x)
2.
log(x – 2) + log(x + 3) = log(x2 + x – 6)
Restrictions:
Restrictions: