Solving Trig Equations Objective: Solve many different Trig equations.
Advanced Algebra with Trig Weeks 5-6 (May 4 May 15) · 2020-04-22 · Complete Study: 11.3.1 Trig...
Transcript of Advanced Algebra with Trig Weeks 5-6 (May 4 May 15) · 2020-04-22 · Complete Study: 11.3.1 Trig...
Advanced Algebra with Trig: Grades 11-12
CHARLES COUNTY PUBLIC SCHOOLS
Advanced Algebra with Trig Mathematics
APEX Learning Packet
Weeks 5-6 (May 4 – May 15)
Advanced Algebra with Trig: Grades 11-12
Student: _________________________________ School: _____________________________
Teacher: _________________________________ Block/Period: ________________________
Packet Directions for Students Week 5:
Read through the Instruction and examples on the 11.3.1 Trig Ratios and the Unit Circle while completing the corresponding questions on the 11.3.1 Study: Trig Ratios and the Unit Circle
Complete Study: 11.3.1 Trig Ratios and the Unit Circle o Check and revise solutions using the Study: 11.3.1 Trig Ratios and the Unit Circle
Answer Key
Note: This content will be assessed on the Quiz at the end of the packet: Trigonometric Functions and the Unit Circle
Week 6:
Read through the Instruction and examples on the 11.3.2 The Pythagorean Theorem while completing the corresponding questions on the 11.3.2 The Pythagorean Theorem
Complete 11.3.2 Study: Pythagorean Theorem o Check and revise solutions using the 11.3.2 Study: Pythagorean Theorem Answer
Key
Complete Quiz: Trigonometric Functions and the Unit Circle
Advanced Algebra with Trig: Grades 11-12
Trigonometric Ratios and the Unit Circle
The trigonometric functions introduced in the last lesson are exactly what we need for modeling this
kind of change. In this lesson, you will see that the six trigonometric functions can be defined using a
unit circle — that is a circle with a radius of 1 — and you will learn to use the unit circle to find values of
the trigonometric functions for angles greater than 90 degrees (or radians).
Terminal Conditions
In this lesson, you will look at the trigonometric functions again, this time from a slightly different
perspective — using a circle. While the triangle allowed us to define the trigonometric functions for
angles between 0 and 90 degrees (or between 0 and radians), the definitions developed in this
lesson will allow us to find values of trigonometric functions for any real number.
We will see that trigonometric functions are especially useful for representing the kind of repetitive
motion seen here as this bicyclist pedals. Notice that her foot goes around and around, repeating the
same motion over and over. The trigonometric functions are sometimes called "circular functions"
because of this repeated circular behavior.
Review the Functions
In the last lesson, you were introduced to the six trigonometric functions and their relationships to the
angles and side lengths of a right triangle.
Advanced Algebra with Trig: Grades 11-12 Trigonometric Functions from the Unit Circle
As useful as trigonometric functions are in relating the sides of a right triangle with its angles, this is not
the only time that these functions prove useful. We will expand our use of trigonometric functions from
angles less than 90 degrees and radians to all possible real angle values.
To define trigonometric functions more generally, begin by looking at the unit circle.
New Definitions for Trigonometric Functions
The table below reviews the new set of definitions for the six trigonometric functions. is the angle (in
radians) determined by the terminal point on the unit circle and can be any real number. The
coordinates of the terminal point are x and y.
Advanced Algebra with Trig: Grades 11-12 The Circle, So Far
The table below summarizes the information you've found so far using the trigonometric definitions
derived from the unit circle. See if you can find any patterns in the values for each function as
angle increases around the entire circle.
(degrees)
(radians)
0 0 0 1 0 undef. 1 undef.
90
1 0 undef. 1 undef. 0
180
0 -1 0 undef. -1 undef.
270
-1 0 undef. -1 undef. 0
You are going to continue to build the unit circle by concentrating on the first quadrant — that is the
part of the circle where x- and y-values are positive. To do this, you can use what you know about the
ratios of the sides of some special right triangles.
Putting it all together, you have the coordinate locations of several more terminal points and their
corresponding angles. This will allow you to solve some trigonometric equations using their definitions.
Take a look at some examples.
Reference Angles
You have begun exploring some new definitions for common trigonometric functions. By now, you've
solved for the coordinates of a few special points on the unit circle that define 30-60-90 and 45-45-
90 triangles in the first quadrant. However, you haven't yet seen how trigonometric functions are
handled when the terminal point is located in quadrants other than the first.
Now you will learn how to use reference angles and reference points to solve for the coordinates of
terminal points on the unit circle located in the second, third, and fourth quadrants.
Advanced Algebra with Trig: Grades 11-12 Reference Angle Examples
The unit circle with reference angles
The Unit Circle from Every Angle
Advanced Algebra with Trig: Grades 11-12
11.3.1 Study: Trig Ratios and the Unit Circle
Name:
Date:
Use the questions below to keep track of key concepts from this lesson's study activity.
Page 1:
Trigonometric functions are sometimes called __________ functions.
Page 2:
Define the six trigonometric ratios for using the triangle below.
a. sin = _______________
b. cos = _______________
c. tan = _______________
d. csc = _______________
e. sec = _______________
f. cot = _______________
Advanced Algebra with Trig: Grades 11-12 Pages 3 – 4:
Define each of the six trigonometric functions when the terminal point P has the coordinates (x,y) in the
unit circle below. Assume x and y are not equal to 0.
a. sin = __________
b. cos = __________
c. tan = __________
d. csc = __________
e. sec = __________
f. cot = __________
Pages 5 – 6:
Fill in the missing information in the table below using the trigonometric definitions derived from the
unit circle.
(degrees) (radians) sin cos tan csc sec cot
0°
180°
Advanced Algebra with Trig: Grades 11-12 Page 7:
Fill in the missing information in the table below using the trigonometric definitions derived from the
unit circle.
(degrees) (radians) sin cos tan csc sec cot
45°
Page 8:
Give the reference angle for each of the following angles.
a.
b.
c.
d.
e.
f.
g.
h.
i.
Pages 9 – 10:
Advanced Algebra with Trig: Grades 11-12 Fill in the missing information in the table below using the trigonometric definitions derived from the
unit circle.
(degrees) (radians) sin cos tan csc sec cot
120°
150°
225°
300°
330°
Advanced Algebra with Trig: Grades 11-12 11.3.1 Study: Trig Ratios and the Unit Circle
ANSWER KEY
Page 1:
Trigonometric functions are sometimes called __________ functions.
circular
Page 2:
Define the six trigonometric ratios for using the triangle below.
a. sin = _______________
b. cos = _______________
c. tan = _______________
d. csc = _______________
e. sec = _______________
f. cot = _______________
Pages 3 – 4:
Advanced Algebra with Trig: Grades 11-12 Define each of the six trigonometric functions when the terminal point P has the coordinates (x,y) in the
unit circle below. Assume x and y are not equal to 0.
a. sin = __________
y
b. cos = __________
x
c. tan = __________
d. csc = __________
e. sec = __________
f. cot = __________
Advanced Algebra with Trig: Grades 11-12 Pages 5 – 6:
Fill in the missing information in the table below using the trigonometric definitions derived from the
unit circle.
The table should appear as follows.
(degrees) (radians) sin cos tan csc sec cot
0° 0 0 1 0 undefined 1 undefined
90°
1 0 undefined 1 undefined 0
180°
0 -1 0 undefined -1 undefined
270°
-1 0 undefined -1 undefined 0
Page 7:
Fill in the missing information in the table below using the trigonometric definitions derived from the
unit circle.
The table should appear as follows.
(degrees) (radians) sin cos tan csc sec cot
30°
2
45°
1
1
60°
2
Advanced Algebra with Trig: Grades 11-12 Page 8:
Give the reference angle for each of the following angles.
a.
b.
c.
d.
e.
f.
g.
h.
i.
Advanced Algebra with Trig: Grades 11-12 Pages 9 – 10:
Fill in the missing information in the table below using the trigonometric definitions derived from the
unit circle.
The table should appear as follows.
(degrees) (radians) sin cos tan csc sec cot
120°
-2
135°
-1
-1
150°
2
210°
-2
225°
1
1
240°
-2
300°
2
315°
-1
-1
330°
-2
Advanced Algebra with Trig: Grades 11-12
11.3.2 Pythagoream Theorem
You can relate the Pythagorean theorem to the unit circle to see a fundamental relationship between
sine and cosine.
Proving It
You could have derived this relationship without using the Pythagorean theorem. How?
The equation of the unit circle is x2 + y2 = 1.
Remember that when we defined cos θ to be x and sin θ to be y, x and y referred to the coordinates of a
point on the unit circle.
What is the equation of the unit circle?
If you substitute cos and sin for x and y in this equation, you obtain the Pythagorean identity:
cos2 θ + sin2 θ = 1
It doesn't matter whether you remember this relationship by thinking about the Pythagorean theorem
or by thinking about the equation of the unit circle. But it does matter that you remember it!
Advanced Algebra with Trig: Grades 11-12
Confirm
You have now learned about unit circles, their trigonometric definitions, and ratios. Answer the question
below to confirm your understanding.
A unit circle is a circle with radius , with the relation between and (x, y) defined
by and .
Definition: Pythagorean identity
Here is one form:
Here are two more:
Example:
(0, -1)
Example: What's the value of the angle with reference point ?
11𝜋
6 (The angle is in the fourth quadrant with a reference angle of )
Example: What is the value of ?
1
2
Example: What is the value of ?
−√3
2 (The sine of the reference angle is . Because the angle of the problem is in the third
quadrant, this value will take a negative sign.)
Advanced Algebra with Trig: Grades 11-12
Here is a summary of what you have seen in this lesson.
Advanced Algebra with Trig: Grades 11-12
11.3.2 Study: Pythagorean Theorem
Study Guide
Name:
Date:
Use the questions below to keep track of key concepts from this lesson's study activity
Pages 1 – 4:
a. What is the equation of the unit circle shown below?
b. What does the Pythagorean theorem say about the relationship between x and y?
c. List the three trigonometric identities that can be derived from the unit circle.
1. ______________________________
2. ______________________________
3. ______________________________
Advanced Algebra with Trig: Grades 11-12
11.3.2 Study: Pythagorean Theorem
ANSWER KEY
Pages 1 – 4:
a. What is the equation of the unit circle shown below?
b. What does the Pythagorean theorem say about the relationship between x and y?
c. List the three trigonometric identities that can be derived from the unit circle.
1. ______________________________
2. ______________________________
3. ______________________________
; ;
Advanced Algebra with Trig: Grades 11-12 Quiz: Trigonometric Functions and the Unit Circle Question 1a of 10
sin( ) = _____
A.
B.
C.
D.
Question 2a of 10
Check all that apply. is the reference angle for:
A.
B.
C.
D.
Advanced Algebra with Trig: Grades 11-12 Question 3a of 10
Which of the following could be points on the unit circle?
A.
B.
C.
D.
Question 4a of 10
If is the point on the unit circle determined by real number , then tan = _____.
A.
B.
C.
D.
Advanced Algebra with Trig: Grades 11-12 Question 5a of 10
If sin > 0 and cos > 0, then the terminal point determined by is in:
A. quadrant 2.
B. quadrant 3.
C. quadrant 1.
D. quadrant 4.
Question 6a of 10
If tan = and the terminal point determined by is in quadrant 3, then:
A.
sin =
B.
csc =
C.
cos =
D.
cot =
Advanced Algebra with Trig: Grades 11-12
7a. The statement "tan = , csc = , and the terminal point determined by is in
quadrant 3":
A. cannot be true because tan is greater than zero in quadrant 3.
B.
cannot be true because if tan = , then csc = .
C. cannot be true because tan must be less than 1.
D. cannot be true because .
Question 8a of 10
Check all that apply. tan is undefined for = _____.
A.
B.
C.
D. 0
Advanced Algebra with Trig: Grades 11-12 Question 9a of 10
sin( ) = _____
A.
B.
C.
D.
Question 10a of 10
cot( ) = _____
A. 0
B. -1
C. 1
D. Undefined