Download Trigonometric Functions Blank Note Packet

Trigonometric Functions 1
6.1 Radian and Degree Measurements
General Set Up and Terms:
Initial Side –
Terminal Side –
Vertex –
Standard Position –
Positive Angles –
Negative Angles –
Coterminal –
What is a RADIAN?
What is the formula for circumference?
What central angle in degrees would go with this?
Plug this in to the formula in the definition of a
radian from the box above to determine how many
radians a whole rotation is.
How can we use this to find how many radians other degrees are?
Trigonometric Functions 2
DO YOUR WORK HERE AND LABEL THE PICTURE ABOVE ACCORDINGLY!
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Let’s use this to complete the formula to convert back and forth
Sketching and Finding Coterminal Angles in both degrees and radians.
Find the coterminal angles for the problems below.
1. The positive angle 135°
2. The positive angle θ =
13 𝜋
3. The Negative angle θ =
SUMMARY OF STEPS:
6
−2 𝜋
3
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Finding Complements and Supplements
Find the complement and supplement of each angle if possible.
1. 72°
2. 148°
3.
4.
2𝜋
5
4𝜋
5
Trigonometric Functions 5
6.2 Trigonometric Functions: The Unit Circle
The Unit Circle Equation:
In short:
Sin 𝜽 =
Cos 𝜽 =
Tan 𝜽 =
𝒔𝒊𝒏 𝜽
𝒄𝒐𝒔 𝜽
=
Trigonometric Functions 6
Evaluating Trig Functions
Step 1: Find the angle on the unit circle.
Step 3: Corresponding (x,y) coordinates on the unit circle.
Step 2: Use the definitions of the trig functions to plug in and simplify.
Examples: Evaluate the six trig functions at each real number.
a. t =
b. t =
𝝅
𝟔
𝟓𝝅
𝟒
c. t = −
𝝅
𝟑
d. t = 
Domain and Range:
To understand the domain and range of sine and cosine, we need to know that it
is:
They repeat values every full rotation:
Trigonometric Functions 7
Looking at our values, which graph is which trig function?
Which ones are odd or even?
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6.3 Trigonometric Functions: The Unit Circle
So you can use these to solve problems that are not on 30°, 45°, 60° and 90° multiples!
Ex. If Cos θ = 0.8, what is Sin θ and Tan θ?
Ex. If Tan θ = 3, what is Cot θ and Sec θ?
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Simplifying using identities
Given 0 < θ<
𝜋
2
transform the left side of the equation to get 1 = 1.
A. cos θ sec θ = 1
Solving Word Problems with Trig
Ex.
B. (sec θ + tan θ) (sec θ – tan θ) = 1
Trigonometric Functions 10
6.4 Trig Functions of any angle
GROUP DISCUSSION:
1. What are the general formulas for Sine, Cosine and Tangent?
2. Now rewrite them with the coordinates (x,y) and the radius on the unit circle.
3. Draw a general triangle with sides labeled (x,y,r) that you would use to show the trig
functions.
4. What formula can you use on the above triangle to find the hypotenuse?
5. What part of the circle is the hypotenuse?
6. On the following Unit Circle, highlight the x-values one color and all the y-values
another.
7. Label each Quadrant with ± x, ± y based on its values.
8. Use your answers in 7, to then put in each Quadrant the sign of that Quadrant’s 3 trig
functions: Sin, Cos and Tan.
9. What are the possible values for the x length, y length and radii? Give each in interval
notation.
10. If you have a negative Sine value for an angle, and a positive Tangent, what quadrant
does your angle have to be in?
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EX. If sinθ =
−2
3
and tan θ > 0, what is the Cos and Cot of that angle?
When an angle’s terminal side is not in Quadrant 1, we need to use the angle’s reference angle.
REFERENCE ANGLE:
Trigonometric Functions 12
There are different formats of these questions, but it all comes back to you old fashioned
GEOMETRY Trig functions.
Group 1: HW Questions 1-3
Group 2: HW Questions 4-6
Group 3: HW Questions 7
Group 4: HW Questions 8-10
Group 5: HW Questions 11
Trigonometric Functions 13
WEBASSIGN PROBLEMS
PROBLEM
1a
1b
2a
2b
3
Coordinates on a triangle Solving for the missing side
Coordinates on a triangle Solving for the missing side
Coordinates on a triangle Solving for the missing side
Coordinates on a triangle Solving for the missing side
Coordinates on a triangle Solving for the missing side
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
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Problem
4
Coordinates on a triangle
*adjust signs based on
constraints
5
6
7
8
Coordinates on a triangle
*adjust signs based on
constraints
Coordinates on a triangle
*adjust signs based on
constraints
Find a
point on
the line.
Where is it
located on
the Circle?
Coordinates on a
triangle
What are the
coordinates there?
Put on a triangle
Solving for the
missing side
Solving for the
missing side
Solving for the
missing side
Solving for the
missing side
Solving for the
missing side
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
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9
10
Where is it
located on
the Circle?
Where is it
located on
the Circle?
What are the
coordinates there?
Put on a triangle
What are the
coordinates there?
Put on a triangle
Solving for the
missing side
Solving for the
missing side
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
6 Trig Functions
Sin =
csc =
Cos =
sin =
Tan =
cot =
11a
At what
angles do
those
coordinates
occur?
What quadrants would
result in that sign of
function?
What values do those
angles or multiples of
that angle occur at?
Convert to radians.
11b
At what
angles do
those
coordinates
occur?
What quadrants would
result in that sign of
function?
What values do those
angles or multiples of
that angle occur at?
Convert to radians.
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6.5 Sine and Cosine Functions
How does the Unit Circle turn into the periodic graphs of sine and cosine??
Cosine: Domain =
Range =
Sine: Domain =
Range
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Now let’s do some translations!!
SINE FUNCTION
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Degrees
0
𝝅
𝟒
𝝅
𝟐
𝟑𝝅
𝟒

𝟓𝝅
𝟒
𝟑𝝅
𝟐
𝟕𝝅
𝟒
2
Range
Domain
Period
Translation
Result
-sin (x)
𝝅
sin (x + )
𝟒
2 + Sin (x)
2 sin (x)
Sin (2x)
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COSINE FUNCTION
Trigonometric Functions 20
Degrees
0
𝝅
𝟒
𝝅
𝟐
𝟑𝝅
𝟒

𝟓𝝅
𝟒
𝟑𝝅
𝟐
𝟕𝝅
𝟒
2
Range
Domain
Period
Translation
Result
cos (-x)
𝝅
cos (x - )
𝟒
cos (x) - 2
𝟏
𝟐
cos (x)
𝟏
cos ( x)
𝟐
Trigonometric Functions 21
General Forms:
y = d + a cos (bx – c)
𝟏
y = d + a sin (bx – c)
𝝅
Ex. Given f(x) = sin (x - ), give the period and critical points within that
𝟐
𝟑
period. Use this information to graph one period of it!!
Amplitude
Period
Where does the period
occur?
= 0
= 2
Split
period
into 4
equal
parts
Find the 5 CRITICAL points
x
y
Trigonometric Functions 22
HOMEWORK: Graph two full periods of the following functions
1. f(x) = 3 sin x
Amplitude
Period
Where does the period
occur?
= 0
= 2
Split
period
into 4
equal
parts
Find the 5 CRITICAL points
x
y
Trigonometric Functions 23
𝟏
2. f(x) = cos x
𝟒
Amplitude
Period
Where does the period
occur?
= 0
= 2
Split
period
into 4
equal
parts
Find the 5 CRITICAL points
x
y
Trigonometric Functions 24
3. f(x) = cos
Amplitude
𝒙
𝟐
Period
Where does the period
occur?
= 0
= 2
Split
period
into 4
equal
parts
Find the 5 CRITICAL points
x
y
Trigonometric Functions 25
𝝅
4. f(x) = sin (x - )
𝟒
Amplitude
Period
Where does the period
occur?
= 0
= 2
Split
period
into 4
equal
parts
Find the 5 CRITICAL points
x
y
Trigonometric Functions 26
5. f(x) = -8 cos (x + )
Amplitude
Period
Where does the period
occur?
= 0
= 2
Split
period
into 4
equal
parts
Find the 5 CRITICAL points
x
y
Trigonometric Functions 27
6.6 Inverse Functions
Let’s recall some general characteristics of an inverse function:
Trigonometric Functions 28
You can calculate the inverse of a trig function just like in Geometry!
You are essential asking yourself…..”At what angle does this trig value occur?”
Examples: Find the exact value of the following problems
Write as a basic
equation
Problem
1. arccos
√2
2
2. 𝑡𝑎𝑛−1 (0)
1
3. arcsin(− )
2
4. 𝑐𝑜𝑠 −1 (−1)
5. arctan 0
6. 𝑠𝑖𝑛−1 (2)
Change the
position of your x
and y isolate the
variable
Solve.
Trigonometric Functions 29
GROUP 1 PROBLEMS
1. The point (−30, 16) is on the terminal side of an angle in standard
position. Determine the exact values of the six trigonometric functions of the
angle.
2. The point (−8, 18) is on the terminal side of an angle in standard
position. Determine the exact values of the six trigonometric functions of the
angle.
3. The point (−10, 22) is on the terminal side of an angle in standard
position. Determine the exact values of the six trigonometric functions of the
angle.
GROUP 2 PROBLEMS
1. Find the values of the six trigonometric functions of θ with the given
constraint.
Function Value
Sin θ =
3
Constraint
In Quadrant II
5
2. Find the values of the six trigonometric functions of θ with the given
constraint.
Function Value
Tan θ =
−15
8
Constraint
Sin θ < 0 AND Tan < 0
3. Find the values of the six trigonometric functions of θ with the given
constraint.
Function Value
tan θ is undefined.
Constraint
 < < 2
Trigonometric Functions 30
GROUP 3 PROBLEMS
1. The terminal side of θ lies on a given line in the specified quadrant. Find
the values of the six trigonometric functions of θ by finding a point on the
line.
Line
Quadrant
40x + 9y = 0
IV
2. The terminal side of θ lies on a given line in the specified quadrant.
Find the values of the six trigonometric functions of θ by finding a point
on the line.
Line
12x + 5y = 0
Quadrant
IV
GROUP 4 PROBLEMS
1. Evaluate the sine, cosine, and tangent of 330°without using a calculator.
2. Evaluate the sine, cosine, and tangent of –
calculator.
3. Evaluate the sine, cosine, and tangent of –
calculator.
11𝜋
6
10𝜋
3
without using a
without using a
Trigonometric Functions 31
GROUP 5 PROBLEMS
1. Find two angles that satisfy the equation.
Give your answers in degrees (0° ≤ θ < 360°) and radians (0 ≤ θ < 2π).
(a) csc(θ) = - √2
(b)
sin(θ) = -1/2
2. Find two angles that satisfy the equation.
Give your answers in degrees (0° ≤ θ < 360°) and radians (0 ≤ θ < 2π).
(a) cot(θ) = –1
(b) sec(θ) = 2
Trigonometric Functions 32
Graphing Sine and Cosine Directions
GRAPH COSINE FROM 0 to 2
1. When graphing a function, which axis is the independent, which is the dependent?
2. Highlight all COSINE values the same color from 0 to 2 .
Graph these using .71 as an estimate for
√2
.
2
NOTE: What is the independent variable here???
GRAPH SINE FROM 0 to 2
3. Highlight all SINE values the same color from 0 to 2 .
Graph these using .71 as an estimate for
√2
.
2
NOTE: What is the independent variable here???
GRAPH COSINE AND SINE FROM 0 to -2
4. What type of function is COSINE?
Use this information to graph your function from 0 to -2
5. What type of function is SINE?
Use this information to graph your function from 0 to -2
To complete your chart:
1. Everyone complete the first column together to ensure we all are completing it properly.
2. Designate a column for each member of your group.
3. Complete the values and graph your function.
4. Use the graphs to fill in your range, domain, period and translation result.
5. When everyone is done, share all your information so that you can get the whole groups results.