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PreCalculus
Section 5-1 Trigonometric Identities
Name:___________________
Basic Trig Identites: See key concept box on page 312
Reciprocal Identities
1
csc 
1
csc  
sin 
sin  
1
sec 
1
sec  
cos 
Quotient Identities
1
cot 
1
cot  
tan 
cos  
tan  
sin 
cos 
cos 
cot  
sin 
tan  
Ex. 1: Use the trig identities to find the trig values.
a. If cos θ = ¾ , find sec θ.
b. If sec x =
5
3
and tan x = , find sin x.
4
4
You can use the unit circle to define trig functions. For any angle θ, sine and cosine are the legs of a right
triangle and the hypotenuse is 1. This leads to the Pythagorean identities. The signs depend on which
quadrant the triangle lies in. See key concept on page 313.
Pythagorean Identities
sin   cos   1
2
2
tan 2   1  sec2 
Ex. 2: a. If cot θ = 2 and cos θ < 0, find sin θ and cos θ.
b. If sin x 
1
, cos x > 0, find cot x and sec x.
6
PreCalculus Ch 5 Notes_Page 1
cot 2   1  csc2 
A trig function f is a cofunction of trig function g if f(α) = g(β) when α and β are complementary angles.
Look at triangle on top of page 314 and the key concept boxes.
Cofunction Identities


sin   cos    
2



tan   cot    
2



sec   csc    
2



cos   sin    
2



cot   tan    
2



csc   sec    
2

Odd  Even Identities
sin      sin 
cos     cos 
tan      tan 
csc      csc 
sec     sec 
cot      cot 
Ex. 4: Use the Odd-Even and Cofunction Identities to find the following.


a. If cos x = -0.75, find sin  x   .
2



b. If sin x = -0.37, find cos  x   .
2

To simplify a trig expression, start by rewriting it in terms of one trig function or in terms of sine and
cosine only.
Ex. 4: Simplify.
1
1  sin 2 x 
a.

cos x
PreCalculus Ch 5 Notes_Page 2
b. sec x – tan x sin x
Ex. 5: Simplify by applying identities and factoring.
a.
cos xtan x – sin xcos2 x
Ex. 6: Simplify by combining fractions.
sec x
sec x

a.
1  sec x 1  sec x
b.
b.


 csc   x   tan 2 x sec x
2

cos x 1  sin x

1  sin x
cos x
Ex. 7: Rewrite the expression as an expression that does not involve a fraction. (Hint: use conjugates)
1  tan 2 x
cos 2 x
a.
b.
csc2 x
1  sin x
PreCalculus Ch 5 Notes_Page 3
PreCalculus
Section 5-2 Verifying Trigonometric Identities
Name:___________________
To verify an identity means to prove that both side of an equation are equal for all values of the variable
for which both sides are defined. Start with the more complicated side and work toward the other side.
You will need state a reason or justification for each step (do a proof).
Ex. 1: Verify each of the following.
tan 2 x  1
a.
 sec4 x
2
1  sin x
b. sec2  cot 2   1  cot 2 
When there are multiple fractions with different denominators, you can find a common denominator to
reduce the expression to one fraction.
Ex. 2: Verify.
2 cot x 
PreCalculus Ch 5 Notes_Page 4
sin x
sin x

1  cos x 1  cos x
When the denominator is in the form 1±u or u±1, use the conjugate of the denominator.
Ex. 3: Verify.
sin x
 cos x cot x  cot x
sec x  1
Until an identity has been verified, you can’t assume that both sides of the equation are equal. Therefore,
you can’t use properties of equality to perform algebraic operations on each side of an identity.
When the more complicated expression of an identity involves powers, try factoring.
Sometimes, it’s helpful to work out each side and meet somewhere in the middle.
Ex. 4: Verify.
a. cos x sec2 x tan x  cos x tan 3 x  sin x
PreCalculus Ch 5 Notes_Page 5
b.
cot 3 x  cot x  cos x csc3 x
* Strategies for verifying trig identities: see Key Concept Summary box on page 323.
 Start with the more complicated side of the identity and work to transform it into the simpler side,
keeping the other side of the identity in mind as your goal.
 Use reciprocal, quotient, Pythagorean, and other basic trig identities.
 Use algebraic operations such as combining fractions, rewriting fractions as sums and differences,
multiplying expressions, or factoring expressions.
 Work each side separately to reach a common intermediate expression.
 If no other strategy presents itself, try converting the entire expression to one involving
only sines and cosines.
Identifying Identities and Nonidentities: You can use a graphing calculator to investigate whether an
equation may be an identity by graphing the functions related to each side of the equation. A graphing
calculator may confirm an identity, but it does not prove an identity. You still need to provide the
algebraic verification of an identity.
Ex. 5: Use a graphing calculator to test whether each equation is an identity. If it appears to be an
identity, verify it. If not, find an x-value for which both side are defined, but not equal.
1  tan 2 x
a.
 tan x
csc x sec x
b.
cos3 x  sin 3 x
 cos 2 x  sin 2 x
cos x  sin x
PreCalculus Ch 5 Notes_Page 6
PreCalculus
Section 5-3 Solving Trigonometric Equations
Name:___________________
See the diagram on page 327 to show that there are an infinite number of solutions for trig functions.
You need to add integer multiples of the period to get all of the solutions. When there is only one trig
expression, start by isolating this expression.
Ex. 1: Solve.
a. 5cos x  3cos x  3
b. 4sin x  2sin x  2
Ex. 2: Solve by taking the square root of each side. (see study tip on page 328)
a. 3 tan 2 x  4  3
b. 3cot 2 x  4  7
PreCalculus Ch 5 Notes_Page 7
You may need to solve by factoring or by using the Quadratic Formula.
Ex. 3: Find all solutions of each equation on the interval [0, 2π).
a. 2sin x cos x  3 sin x
b. 2sin 2 x  sin x  1  0
When equations involve multiple angles, such as cos 2x= 1, first solve for the multiple angle 2x.
Ex. 4: A projectile is sent off with an initial speed v0 of 350m/s and clears a fence 3000m away. The
height of the fence is the same height as the initial height of the projectile. If the distance the projectile
v 2 sin 2
traveled is given by d  0
, find the interval of possible launch angles to clear the fence.
9.8
PreCalculus Ch 5 Notes_Page 8
Ex. 5: Find all solutions of sin 2 x  sin x  1  cos 2 x on the interval [0, 2π).
Sometimes you can square both sides of the equation, but this may produce extraneous solutions.
So make sure to check for the extraneous solutions when you squared the original equation.
Ex. 6: Find all solutions of sin x  cos x  1 on the interval [0, 2π).
PreCalculus Ch 5 Notes_Page 9
PreCalculus
Sections 5-4 Sum and Difference Identities
Name:___________________
Look at page 336 to see the Cosine of a Sum and Difference Identities. See the key concept on page 337.
Sum Identities
Difference Identities
cos      cos  cos   sin  sin 
cos      cos  cos   sin  sin 
sin      sin  cos   cos  sin 
sin      sin  cos   cos  sin 
tan     
tan   tan 
1  tan  tan 
tan     
tan   tan 
1  tan  tan 
You can rewrite angle measures as the sums and differences of special angles and use these identities to
find exact values of trig functions of angles that are less common.
Ex. 1: Find the exact value of each trig expression.
a. cos 75˚
b. tan
11
12
Ex. 2: An alternating current i in amperes in a certain circuit can be found after t seconds using
i= 4(sin255)t. (a) Rewrite the formula in terms of the sum of two angle measures.
(b) Use a sine of a sum identity to find the exact current after 1 second.
PreCalculus Ch 5 Notes_Page 10
If a trig expression has the form of a sum or difference identity, you can use the identity to find the exact
value or to simplify an expression by rewriting the expression as a function of a single angle.
Ex. 3: a. Find the exact value of
tan 78  tan18
1  tan 78 tan18
b. Simplify: sin

3
cos

4
 cos

3
sin

Sum and Difference Identities can be used to rewrite trig expressions as algebraic expressions.


3
 arccos x  as an algebraic expression of x that does not involve trig
Ex. 4: Write cos  arcsin
2


functions.
Ex. 5: Verify:
cos     cos
PreCalculus Ch 5 Notes_Page 11
4
Sum and difference identities can be used to rewrite trig expressions in which one of the angles is a
multiple of 90˚ or π/2 radians. The resulting identity is called a reduction identity because it reduces the
complexity of the expression.
Ex. 6: Verify each reduction identity.


a. cos      sin 
2

b. tan  x  360  tan x




Ex. 7: Find the solutions of sin  x    sin  x    0 on the interval [0, 2π).
4
4


PreCalculus Ch 5 Notes_Page 12
PreCalculus
Section 5-5 Multiple Angle Identity
Name:_________________________
By letting α and β both equal θ in each of the angle sum identities, you can derive the following
double-angle identities.
Double  Angle Identities
sin 2  2sin  cos 
tan 2 
2 tan 
1  tan 2 
cos 2  cos 2   sin 2 
cos 2  2 cos 2   1
cos 2  1  2sin 2 
Ex. 1: a. If sin  
3
 
on the interval  0,  , find sin2θ, cos2θ, and tan2θ.
4
 2
3
 
b. If cos   on the interval  0,  , find sin2θ, cos2θ, and tan2θ.
5
 2
Ex. 2: Solve cos2θ – cosθ = 2 on the interval [0, 2π).
PreCalculus Ch 5 Notes_Page 13
The double-angle identities can be used to derive the following power-reducing identities.
Power  Reducing Identities
sin 2  
1  cos 2
2
cos 2  
1  cos 2
2
tan 2  
1  cos 2
1  cos 2
Ex. 3: Rewrite the trig function in terms of cosines of multiple angles with no power greater than 1.
a. csc 4 
b. cos 4 
Ex. 4: Solve sin 2   cos 2  cos   0
PreCalculus Ch 5 Notes_Page 14

in each of the power-reducing identities, you can derive the half-angle identities.
2

The sign of the identity that involves the ± is determined by which quadrant the terminal side of
lies.
By replacing θ with
2
Half  Angle Identities
sin
tan

2


2
1  cos 
1  cos 

1  cos 
2
tan
cos

2

1  cos 
sin 
Ex. 5: Find the exact values.
a. sin 22.5˚
b. tan

2

1  cos 
2
tan

2
7
12
Ex. 6: Solve the trig equations on the interval [0, 2π].
x
x
x
a. sin 2  cos 2
b. 2sin 2  cos x  1  sin x
2
2
2
PreCalculus Ch 5 Notes_Page 15

sin 
1  cos 