Download Syllabus Objectives: 3

Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
4.7
Inverse Trig Functions
Syllabus Objective: 4.2 – The student will sketch the graphs of the principal inverses of the six
trigonometric functions.
Recall: In order for a function to have an inverse function, it must be one-to-one (must pass both the
horizontal and vertical line tests).
Notation: The inverse of f  x  is labeled as f 1  x  .
y  sin x
Graph of f  x   sin x Domain:
Range:
  
In order for f  x   sin x to have an inverse function, we must restrict its domain to   ,  .
 2 2
Inverse of the Sine Function
To graph the inverse of sine, reflect about
the line y  x .
Domain of f 1  x   sin 1 x :
Notation: Inverse of Sine
Range of f 1  x   sin 1 x :
f 1  x   sin 1 x or y  arcsin x (arcsine)
Note: y  sin1 x denotes the inverse of sine (arcsine). It is NOT the reciprocal of sine (cosecant).
Page 1 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Evaluating the Inverse Sine Function
Ex1: Find the exact values of the following.
1.
arcsin
2
2
What value of x makes the equation sin x 
2
true?
2
  
Note: The range of arcsine is restricted to   ,  , so _____ is the only possible answer.
 2 2
1
2. sin 3
What value of x makes the equation sin x  3 true? ____________________
3.
 2 
sin 1  sin
 Taking the inverse sine of the sine function results in the argument.
3 

Inverse of the Cosine Function
Graph of f  x   cos x
Domain:
Range:
In order for f  x   cos x to have an
inverse function, we must restrict its
domain to 0,  .
To graph the inverse of cosine, reflect about the line y  x .
Domain of f 1  x   cos1 x :
Notation: Inverse of Cosine
Range of f 1  x   cos1 x :
f 1  x   cos1 x or y  arccos x (arccosine)
Note: y  cos1 x denotes the inverse of cosine (arccosine). It is NOT the reciprocal of cosine
(secant).
Page 2 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Evaluating the Inverse Cosine Function
Ex2: Find the exact values of the following.
1.

2
2
arccos  
true?
 What value of x makes the equation cos x  
2
 2 
Note: The range of arcsine is restricted to 0,  , so ________ is the only possible answer.
2.

 3 
sin  cos1 
 

 2 

3.
11 

cos1  cos

6 

 3
cos 1 
 
 2 
, so
 
sin   
6
Inverse of the Tangent Function
Graph of f  x   tan x
Domain:
Range:
In order for f  x   tan x to have an inverse
function, we must restrict its domain to
  
  2 , 2  .
To graph the inverse of tangent,
reflect about the line y  x .
Domain of f 1  x   tan 1 x :
Notation: Inverse of Tangent
Range of f 1  x   tan 1 x :
f 1  x   tan 1 x or y  arctan x (arctangent)
Note: y  tan 1 x denotes the inverse of tangent (arctangent). It is NOT the reciprocal of tangent
(cotangent).
Page 3 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Evaluating the Inverse Tangent Function
Ex3: Find the exact values of the following.
1.

3
sin  tan 1

3 


3
sin  tan 1
  sin________  ________
3 

3
  
Note: The range of arctangent is restricted to   ,  , so ___ is the only possible answer for tan 1
.
3
 2 2
2.
cos  tan 1 1
cos  tan 1 1  cos______  ________
3.


arccos  tan 
3



arccos  tan   arccos______ No Solution, because _______
3

Right Triangle Trigonometry and Inverse Trigonometric Functions: the trigonometric functions can be
evaluated without having to find the angle
 Label the sides of the right triangle based upon the inverse trig function given
 Evaluate the length of the missing side (Pythagorean Theorem)
 Evaluate the trig function – be sure to choose the correct sign!
1

Ex4: Evaluate cos  arctan  without a calculator.
5

θ
Right Triangle
Hypotenuse:
1
  
Let   arctan . Since the range of arctangent is   ,  , and the tangent is positive,  must be in
5
 2 2
Quadrant ______. Therefore, cosine is positive. So cos  .
Ex5: Find an algebraic expression equivalent to sin arccos  4x  .
θ

  
You Try: Evaluate cos 1  cos     . Be careful!
 4 

QOD: Explain how the domains of sine, cosine, and tangent must be restricted in order to create an
inverse function for each.
Page 4 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
4.8
Trig Application Problems
Syllabus Objective: 4.5 – The student will model real-world application problems involving graphs
of trigonometric functions.
Angle of Elevation: the angle through which the eye moves up from horizontal to look at something
above
Angle of Depression: the angle through which the eye moves down from horizontal to look at something
below
Angle of Elevation
Angle of Depression
Solving Application Problems with Trigonometry:
 Draw and label a diagram (Note: Diagrams shown are not drawn to scale.)
 Find a right triangle involved and write an equation using a trigonometric function
 Solve for the variable in the equation
Note: Be sure your calculator is in the correct Mode (degrees/radians).
Ex1: If you stand 12 feet from a statue, the angle of elevation to the top is 30°, and the angle of
depression to the bottom is 15°. How tall is the statue?
Height of the statue is approximately
Ex2: Two boats lie in a straight line with the base of a cliff 21 meters above the water. The angles of
depression are 53° to the nearest boat and 27° to the farthest boat. How far apart are the boats?
Distance between the boats is approximately
Page 5 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Ex3: A boat leaves San Diego at 30 knots (nautical mph) on a course of 200°. Two hours l
ater the boat changes course to 290° for an hour. What is the boat’s bearing and distance from San
Diego? Remember: bearing starts N, clockwise
Simple Harmonic Motion: describes the motion of objects that oscillate, vibrate, or rotate; can be
modeled by the equations d  a sin  bt  or d  a cos  bt  .
Frequency =
b
; the number of oscillations per unit of time
2
Ex4: A mass on a spring oscillates back and forth and completes one cycle in 3 seconds. Its
maximum displacement is 8 cm. Write an equation that models this motion.
Period =
Amplitude =
You Try: You observe a rocket launch from 2 miles away. In 4 seconds, the angle of elevation changes
from 3.5° to 41°. How far did the rocket travel and how fast?
QOD: What is the difference between an angle of depress
Page 6 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
5.1
Using Fundamental Identities
+
Syllabus Objectives: 3.3 – The student will simplify trigonometric expressions and prove
trigonometric identities (fundamental identities). 3.4 – The student will solve trigonometric
equations with and without technology.
Identity: a statement that is true for all values for which both sides are defined
Example from algebra: 3 x  8  11  3x  13
Simplifying Trigonometric Expressions:
 Look for identities
 Change everything to sine and cosine and reduce. Eliminate fractions.
 Algebra: mulitiply, factor, cancel….
Ex1: Use basic identities to simplify the expressions.
a)

cot  1  cos2 

sin2   cos2   1  sin2   1  cos2 
b) tan   csc 
Ex2: a. Simplify the expression (sin x – 1)(sin x + 1)
b. Simplify the expression
 csc x  1 csc x  1 .
cos2 x
1  cot 2   csc 2   cot 2   csc 2   1
Use algebra:
Ex3: a. Simplify the expression sin   x  csc   x  .
sin   x    sin x csc   x    csc x
b. cos (θ – 90°)
Simplifying Trigonometric Expressions: Simplify using the following strategies. Note that the equations
in bold are the trig identities used when simplifying. All of the other steps are algebra steps.
Ex4: Simplify the expression by factoring. sin2 x  cos2 x  1
a.
b.

cos3 x  cos x sin 2 x

csc2 x  cot x  3
Page 7 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
c.
sec2   1
d.
4 tan 2   tan   3
sin x
cos x
Ex5: Simplify the expression by combining fractions.

1  cos x sin x
sin2 x  cos2 x  1
csc x 
1
sin x
Verify numerically, graphically.
Ex. 6 Rewrite
1
so that it is not in fractional form by Multiplying by the conjugate.
1  sin x
Ex 7: Verify the Trigonometric Identity. (numerically, graphically)
cos 3x  4 cos 2 x  3cos x
Ex. 8: Use x  2 tan  , 0   

2
, to write
4  x 2 as a trigonometric function of 
Reflection:
Page 8 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
5.2
Verify Trigonmetric Identities
Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove
trigonometric identities.
Trigonometric Identity: an equation involving trigonometric functions that is a true equation for all
values of x
Tips for Proving Trigonometric Identities: (We are not solving. Do not do anything to both sides.)
1. Manipulate only one side of the equation. Start with the more complicated side.
2. Look for any identities (use all that you have learned so far).
3. Change everything to sine or cosine.
4. Use algebra (common denominators, factoring, etc) to simplify.
5. Each step should have one change only.
6. The final step should have the same expression on both sides of the equation.
Note: Your goal when proving a trig identity is to make both sides look identical!
For all of the following examples, prove that the identity is true. The trig identities used in the
substitutions are in bold.


Ex1:
cos3 x  1  sin 2 x cos x
Ex2:
1
1

 2sec2 x
1  sin x 1  sin x
Start with the right side (more complicated).
sin2 x  cos2 x  1  cos2 x  1  sin2 x
Start with the left side.
Combine fractions.
Simplify.
Trig substitution.
sin2 x  cos2 x  1  cos2 x  1  sin2 x
Identity
1
 sec x
cos x
Page 9 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Ex3:
 tan
2


x  1 cos2 x  1   tan 2 x
Start with the left side.
Trig substitution.
tan 2   1  sec 2 
Trig substitution.
sin2 x  cos2 x  1  cos2 x  1   sin 2 x
Trig substitution
1
 sec x
cos x
Multiply.
Identitiy.
sin x
 tan x
cos x
Ex4: sec x  tan x 
cos x
1  sin x
Start with the left side.
Change to sine/cosine.
Combine fractions.
Multiply num/den by conjugate.
Trig substitution.
sin2 x  cos2 x  1  cos2 x  1  sin2 x
Simplify.
Ex5:
sec 2   1
 sin 2 
2
sec 
Start with left side.
Split the fraction.
Simplify.
Trig substitution.
sin2   cos2   1  sin2   1  cos2 
Identity.
1
 cos
sec 
Challenge: Try to prove the identity above in another way.
You Try: Prove the identity.
 cos x  sin x 
2
  cos x  sin x   2
2
Reflection: List at least 5 strategies you can use when proving trigonometric identities.
Page 10 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
5.3
Solving Trigonmetric Identities
Solving Trigonometric Equations
 Isolate the trigonometric function.
 Solve for x using inverse trig functions. Note – There may be more than one solution or no
solution.
Ex1: Solve the equation 4sin 2 x  4  0 in the interval 0,2  .
Find values of x for which x  sin 1 1 and x  sin 1  1 :
x
Solving Trigonometric Equations: Solve using the following strategies. Find all solutions for each
equation in the interval 0,2  .
Ex2: Solve the equation by isolating the trig function. 2cos x  1  0
These are values of x where the cosine is equal to
1
.
2
Ex3: Solve the equation by extracting square roots. 4sin 2 x  3  0
These are values of x where the sine is equal to 
Page 11 of 22
3
.
2
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Ex4: Solve the equation by factoring. 2cos2 x  cos x  1
Set equal to zero.
Factor.
Set each factor equal to zero.
Solve each equation.
Note: It may be easier to use u-substitution with u  cos x to help students visualize the equation as a
quadratic equation that can be factored.
Ex5: Solve the equation by factoring. 2sec x sin x  sec x  0
Factor out GCF.
Use zero product property.
Solve each equation.
Note: It is possible for an equation to have no solution.
2
Ex6: Solve by rewriting in a single trig function. 2sin x  3cos x  3
Substitute Pyth. Identity.
sin2   cos2   1  sin2   1  cos2 
Simplify algebraically.
Factor and solve.
Page 12 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Ex7: Solve using trig substitutions.
sin 3 x
 tan x
cos x
Rewrite sin 3 x  sin 2 x sin x
Rewrite
sin x
 tan x .
cos x
Ex. 8 Solve the Function of a multiple angle.
2cos3t 1  0
1. First solve for 3t
2. Then divide the results by 3
Ex9: Find the approximate solution using the calculator. 4 cos x  1
1
Isolate the trig function.
cos x 
4
1
To find x, we need to find the inverse cosine of ¼. x  cos1  
4
x  ____
When solving an equation in the interval 0,2  , be sure to be in Radian mode.
You Try: Make the suggested trigonometric substitution and then use the Pythagorean Identities to write
the resulting function as a multiple of a basic trig function.
4  x 2 , x  2cos
Reflection: Explain the relationship between trig functions and their cofunctions.
Page 13 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
5.4
Sum and Difference Formulas
Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove
trigonometric identities (sum and difference identities).
Recall:
36  64  100  10
So in general,
36  64  36  64  6  8  14
ab  a  b
and
 2  32  52  25
 2  32   22  32  13
So in general, (a  b)2  a 2  b2
Sum and Difference Identities
sin  u  v   sin u cos v  cos u sin v
cos  u  v   cos u cos v sin u sin v
tan  u  v  
tan u  tan v
1 tan u tan v
Note: Be careful with +/− signs!
Simplifying Expressions with Sum and Differences
1. Rewrite the expression using a sum/difference identity.
2. Simplify the expression and evaluate if necessary.
Ex1: Write the expression as the sine of an angle. Then give the exact value.




sin cos  cos sin
4
12
4 12
sin  u  v   sin u cos v  cos u sin v
Evaluating Trigonometric Expressions with Non-Special Angles
1. Rewrite the angle as a sum or difference of two special angles.
2. Rewrite the expression using a sum/difference identity.
3. Evaluate the expression.
or
Ex2: Find the exact value of cos195 .
195  150  45
cos195 
cos  u  v   cos u cos v  sin u sin v
Page 14 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Ex3: Write as one trig function and find an exact value.
tan  u  v  
tan80  tan 55
1  tan80 tan 55
tan u  tan v
1  tan u tan v
Evaluating Trig Functions Given Other Trig Function(s)
15
3
4

,  u
and sin v  , 0  v  .
17
2
5
2
We must find cosv and sin u .
Ex4: Find cos  u  v  given cos u  
cos  u  v   cos u cos v  sin u sin v
Draw the appropriate right triangles in the coordinate plane.
15
3
4

:
cos u   ,   u 
sin v  , 0  v  :
17
2
5
2
15
u
5
4
17
v
Use the Pythagorean Theorem to find the missing sides.
In Quadrant III, sine is negative, so sin u  _____ . In Quadrant I, cosine is positive, so cos v  _____ .
4
15
sin v 
17
5
cos  u  v   cos u cos v  sin u sin v 
cos u  
Page 15 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Proving Identities
Ex5: Verify the identity.
sin    
 tan   tan 
cos  cos 
Start with the left side.
Trig substitution:
sin  u  v   sin u cos v  cos u sin v
Split the fraction:
Simplify:
Trig substitution:


You Try: Verify the cofunction identity sin      cos using the angle difference identity.
2

Reflection: Give an example of a function for which f  a  b   f  a   f  b  for all real numbers a and
b. Then give an example of a function for which f  a  b   f  a   f  b  for all real numbers a and b.
Page 16 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Date:
5.5
Multiple Angles
Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove
trigonometric identities (double angle and power-reducing identities).
Ex1: Derive the double angle identities using the sum identities.
a.) sin  2u   sin  u  u 
b.) cos  2u   cos  u  u 
c.) tan  2u   tan  u  u 
Double Angle Identities
sin 2  2sin  cos
cos 2  cos 2   sin 2 
tan 2 
2 tan 
1  tan 2 
There are two other ways to write the double angle identity for cosine. Use the Pythagorean
identity.
sin 2   cos 2   1
cos 2  cos2   sin 2 
sin 2   1  cos 2 
cos 2  1  2sin 2 
cos 2   1  sin 2 
cos 2  2cos2   1
Page 17 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Evaluating Double-Angle Trigonometric Functions
Ex2: Find the exact value of cos 2u given cot u  5,
3
 u  2 .
2
5
u
1
u will be in Quadrant IV and forms a right triangle as labeled.
Using the Pythagorean Theorem, we have
5
1
cos u 
, sin u  
Double Angle Identity: cos 2u  cos2 u  sin 2 u
26
26
Note: If u is in Quadrant IV,
3
 u  2 , then for 2u we have
2
which is in Quadrant IV. So it makes sense that cos 2u is positive.
Solving Trigonometric Equations
Ex3: Find the solutions to 4sin x  cos x  1 in 0,2  .
Rewrite the equation.
Trig substitution.
sin 2 x  2sin x cos x
Isolate trig function.
Solve for the argument.
Because the argument is 2x, we must revisit the domain. 0,2  is the restriction for x. So
0  x  2 . Therefore,.
Solve for x.
Page 18 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Rewriting a Multiple Angle Trig Function to a Single Angle
Ex4: Express sin 3x in terms of sin x .
Rewrite argument as a sum
Sum identity
Double angle identities
Pythagorean identity
Simplify
Verifying a Trig Identity
Ex5: Verify sin 2 
2 tan 
.
1  tan 2 
Start with left side.
Pythagorean identity
Rewrite in sines/cosines
Simplify
Double angle identity
Solving for sin 2  and cos2  , we can derive the power reducing identities.
cos2  1  2sin 2 
sin 2  
1  cos2
2
cos2  2cos2   1
cos2  
1  cos2
2
1  cos 2
sin 2 
1  cos 2
2
tan  


2
1

cos
2

1  cos 2
cos 
2
2
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Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Power Reducing Identities
1  cos 2
2
1  cos 2
cos 2  
2
1

cos
2
tan 2  
1  cos 2
sin 2  
Ex6: Express cos5 x in terms of trig functions with no power greater than 1.
Rewrite as a product
Power reducing identity
Multiply
Power reducing identity
You Try:
1. Find the solutions to 2cos x  sin 2 x  0 in 0,2  .
2. Verify
2cos2
 cot   tan  .
sin 2
Reflection: How do you convert from a cosine function to a sine function? Explain.
Page 20 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
5.5
Half-Angle Identities
Syllabus Objective: 3.3 – The student will simplify trigonometric expressions and prove
trigonometric identities (half angle identities).
Recall: sin2  
1  cos 2
2
Let  
u
 u  1  cos u
. We have sin 2   
2
2
2
1  cos u
u
u
Solving for sin   , we have sin    
. All of the other half-angle identities can be derived
2
2
2
in a similar manner.
Half-Angle Identities
sin
u
1  cos u

2
2
u
1  cos u
cos  
2
2
tan
u 1  cos u

2
sin u
u
sin u
tan 
2 1  cos u
tan
Note: There are 2 others for tangent.
u
1  cos u

2
1  cos u
Note: The  will be decided based upon which quadrant
u
lies in.
2
Evaluating Trig Functions
Ex1: Find the exact value of
a.) cos

12
Rewrite as a half angle
Half angle identity

12
is in Quadrant I, where cosine is positive.
Evaluate
Choose sign
b.) tan22.5̊
Page 21 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4
Precalculus Notes: Unit 4.7-4.8 & 5 – Trigonometric Identities
Solving a Trig Equation
Ex2: Solve the equation sin x  sin
x
in 0,2  .
2
Half-angle identity
Square both sides
Pythagorean identity
Set equal to zero
Factor
Zero product property
You Try:
Solve: sin2
x
2
+ cos x = 0
Reflection: Explain why two of the half-angle identities do not have +/− signs.
Page 22 of 22
Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4