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Unit 1: Trigonometric Functions
SQUARE ROOT REVIEW
Perfect Squares:
5
exact
form
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, …
Simplified Form
 2.236067977...
approximate form
If a square root has a factor that is a perfect square, then it can
be simplified.
Ex.
24
Rationalizing the Denominator
Multiplication Property of Square Roots
The objective is to remove the square root from the
denominator.
a  b  a b
m a n b  m n a b
Ex.
7
2
Ex. 2 3  5 2
Ex.
9
3
Ex.
5
Ex. 2  3
Prove the 30-60-90 Triangle Relationships.
2 3
Prove the 45-45-90 Triangle Relationships.
Unit 1: Trigonometric Functions
Lesson 1 – Angles

I can know and understand basic angle terminology, including that a positive angle has counterclockwise rotation and a negative angle has
clockwise rotation

I can understand what it means for an angle to be measured in degrees

I can use both decimal degrees and DMS angles and convert between forms

I can draw an angle in standard position and know terminology such as quadrantal angles, coterminal angles, reference angles.
Basic Angle Terminology
-
Angle formed by
-
Label the initial side, terminal side, and vertex.
-
Angle is positive if the rotation is
.
-
Angle is negative if the rotation is
.
Degree Measure
-
The degree is the most common unit for measuring angles.
-
A complete rotation of a ray gives an angle whose measure is 360 degrees.
-
1 degree =
-
Complementary Angles sum to 90 degrees; Supplementary Angles sum to 180 degrees
Decimal Degrees and “DMS” Form
When an angle has a measure with a portion of a degree, such as 45.34 degrees, the portion is divided
into “minutes” and “seconds”.
A minute (1’) is 1/60th of a degree
A second (1”) is 1/60th of a minute
Convert 45.34 into DMS form.
IN OTHER WORDS…
Convert 12 42'38 " into decimal degrees.
Unit 1: Trigonometric Functions
Ex.
Perform each calculation without a calculator.
1. 22 13' 81 41'
2. 56 71' 104 39 '
3. 90  51 28'
Your calculator, by default will put an angle in decimal form. You can use your calculator to convert back
and forth.
Standard Position
An angle in standard position is an angle with
.
A standard position angle whose terminal side is ON the x- or y-axis is called a
.
Coterminal Angles
When two angles have same initial side and same terminal side, but different amounts of rotation, they
are called coterminal angles.
Examples:
Coterminal angles are found by
Ex. What is the smallest positive angle coterminal with 1789 ?
Ex. What is the smallest positive angle coterminal with 700 ?
.
Unit 1: Trigonometric Functions
Lesson 2 – Angles in Standard Position

I can write the six trigonometric functions of an angle in standard position given a point on its terminal side

I can write the six trigonometric functions of an angle in standard position, given the equation of the line through the ray

I can find the value of a trigonometric function of a quadrantal angle.
Definition of the Trigonometric Functions
Ex:
Draw the angle  in standard position having the point (9, 12) on its terminal side. Write the six
trigonometric functions of the angle.
Ex:
Draw the angle  in standard position having the point (-6,-1) on its terminal side. Write the six
trigonometric functions of the angle.
Unit 1: Trigonometric Functions
Ex:
Write the six trigonometric function values of the angle  in standard position, if the terminal side
of  is defined by the equation 4 x  7 y  0 , and x  0 .
Quadrantal Angles
Ex. Find the values of the six trigonometric functions for:
1. an angle of 180 degrees
2. an angle  in standard position with terminal
side through (0,-4)
Unit 1: Trigonometric Functions
Lesson 3 – ASTC, Ranges of the Trigonometric Functions

I can determine if the value of a trigonometric function of an angle in standard position is positive or negative

I can know the ranges of the six trigonometric functions and explain why they are what they are
ASTC
Using the definition of the trigonometric functions, determine where each trig function is positive or
negative.
Ex.
Identify the quadrant(s) of any angle  that satisfies sin  0 and cos  0 .
Identify the quadrant(s) of any angle  in which tan  0 .
Ex.
Determine if the following are positive or negative.
sin99
tan300
sec 400
Unit 1: Trigonometric Functions
Ranges of the Trigonometric Functions:
Before you determine what the ranges are for the six trigonometric functions, think about any restrictions
on the values of x, y, and r.
Sine:
Domain:
Range:
Cosine:
Domain:
Range:
Tangent:
Domain:
Range:
Cosecant:
Domain:
Range:
Secant:
Domain:
Range:
Cotangent:
Domain:
Range:
Ex. Determine if each has a defined value or is undefined.
cos90
sec 90
tan180
cot 0
Ex. Determine if the following statements are possible or impossible. Explain your answer.
sin  .245
tan  0
sec   1.4
cos  
5
3
Unit 1: Trigonometric Functions
Ex.
Suppose that an angle  is in quadrant II and sin 
5
. Find the values of the other five
7
trigonometric functions using the definitions of the trigonometric functions.
Ex.
Suppose that an angle  is in quadrant III and cot  
4
. Find the values of the other five
5
trigonometric functions using the definitions of the trigonometric functions.
Unit 1: Trigonometric Functions
Lesson 4 – The Fundamental Identities

I can derive and use the basic trigonometric identities.
Using the definitions of the trigonometric functions, you can derive the fundamental trigonometric
identities.
What is an identity?
Reciprocal Identities
Pythagorean Identities
Starting with the relationship:
x2  y 2  r 2
Divide by r 2
Divide by x 2
Quotient Identities
Divide by y 2
Unit 1: Trigonometric Functions
Complete the following examples using the fundamental identities.
Be sure that you have work to verify your answer, showing all steps.
Ex.
sin 

Show that if sec  
3
, then
2
5
, with  in quadrant I.
3
Ex.
Show that if csc   6 , then
tan  
35
, with  in quadrant IV.
35
Unit 1: Trigonometric Functions
Lesson 5 – Trig Functions of Acute Angles

I can know and use the right triangle definitions of the trigonometric functions

I can know which trigonometric functions are cofunctions with each other

I can find trigonometric function values of 30, 45, and 60 degree angles
In the last chapter, we used angles in standard position to define the trigonometric functions (x, y, and r).
Trig functions are also defined for acute angles of right triangles.
Right Triangle-Based Definitions of Trig Functions
Ex.
Find the exact values of the trigonometric functions of angle A.
B
26
24
Find the exact values of the trigonometric functions of angle B.
A
Cofunctions
In a right triangle, A and B are always complementary.
For any acute angle A,
Ex.
Write Functions in Terms of Cofunctions
1. sec 39
2. sin18.7
3. tan30
Unit 1: Trigonometric Functions
Special Right Triangles
Recall the relationships between the sides of a 30-60-90 triangle and a 45-45-90 triangle.
Using the right-triangle based definitions and these special triangles, you can find the value of a trig
function of these special angles: 30 , 45 , and 60 .
Ex. sin60
=
Ex. sec 45 =
Unit 1: Trigonometric Functions
Lesson 6 – Trig Functions of Non-Acute Angles

I can draw an angle in standard position and know terminology such as quadrantal angles, coterminal angles, reference angles

I can find trigonometric function values of angles with 30, 45, and 60 degree reference angles.
Reference Angles
A reference angle is the positive acute angle made by the terminal side of an angle and the x-axis.
Ex. What is the reference angle for the angle 250 ?
We can find the exact trigonometric function values for any angle with a 30, 45, or 60 degree reference
angle.
To find the value,
1. Determine the quadrant the angle is in, and whether that trig function is positive or negative in that
quadrant.
2. Determine reference angle, and determine the value for that trig function for that reference angle.
Ex.
sin210
Find the exact value of each without a calculator.
csc 300
tan150
cot 225
Unit 1: Trigonometric Functions
Lesson 7 – The Unit Circle

I can know, understand, and use the unit circle with degree angles .
The unit circle organizes the angles in standard position that have 30, 45, and 60 degree reference
angles as well as your quadrantal angles. The radius of a unit circle is
unit.
Unit 1: Trigonometric Functions
Because the sine ratio is opposite over hypotenuse (which is 1 on a unit circle), the sine of an angle is
the
.
Likewise, because the cosine ratio is the adjacent over hypotenuse, the cosine of an angle is the
.
Because the tangent ratio is
sine
, the tangent of an angle is
cosine
.
The cosecant, secant, and cotangent functions are reciprocals of sine, cosine, and cotangent functions,
respectively.
Ex. Find the value of the following. Do not use a calculator.
a. sin225
b. cos150
c. tan 480
d. csc 315
e. sec 210
f. cot 300
Ex. Find all values of  , if  is in the interval 0 ,360
a. cos   
2
2
 . No calculator.
b. sec   
2 3
3
c. cot    3
Unit 1: Trigonometric Functions
Lesson 8 – The Inverse Trigonometric Functions (Introduction)

I can understand why the inverse trigonometric functions have restricted ranges

I can know the domains and restricted ranges for the inverse trigonometric functions

I can evaluate expressions involving inverse trigonometric functions without a calculator

I can evaluate expressions involving inverse trigonometric functions with a calculator
The inverse trigonometric functions are the inverse functions of the trigonometric functions, which means
that the instead the domain being angles in degrees, the ranges are now angles in degrees.
For example,
But, this is tricky since we have seen that trig functions of multiple angles can have the same output.
For example,
For this reason, the inverse trigonometric functions have restricted ranges, meaning that their output is
restricted to certain angles.
Using your calculator, determine what you should expect when you type the following on your calculator.
Range
Function
Domain
Positive
Negative
sin1  ____ 
cos1  ____ 
tan1  ____ 
Without your calculator,
Ex. The output for the expression sin1  .4568902 should be an angle in the range
.
Ex. The output for the expression cos1  .1289555  should be an angle in the range
.
Ex. The output for the expression tan1 .4568902 should be an angle in the range
.
Unit 1: Trigonometric Functions
Lesson 9 – Trig Functions on the Calculator

I can find trigonometric function values of degree angles using a calculator

I can find the angle(s) that have a particular trigonometric function value using a calculator.
*For the time being, be sure to make sure your calculator is in degree mode.
Ex. Find approximate values of trig functions. Round to eight decimal places.
sin35 14'
cos(10 )
tan735.63
There’s no button! How could use the other trigonometric functions?
csc 56
sec 95 15'3"
cot(56 )
When given the value of a trig function, you may use the inverse functions ( sin1 , cos 1 , tan1 ).
Ex.
Find all values of  , if  is in the interval [0 ,360 ] that satisfies each condition.
Leave answers in decimal degrees.
Round to the nearest hundredth of a degree.
a. sin  .88888
b. tan  .98765422
Unit 1: Trigonometric Functions
Again, there is no csc 1 , sec 1 , or cot 1 function on your calculator. How could you use the other
inverse functions?
c. csc   1.2345678
d. sec   .3645
e. cot   9.11
f. sec   2.5