ALGEBRA 2 X
|DAY |DATE |TOPIC |ASSIGNMENT |
|1 | |13.1 Trigonometric Functions Introduction |Day 1: Worksheet A |
|2 | |13.1 Special Right Triangles |Day 2: Worksheet B |
|3 | |13.1 Solving Right Triangles: Finding missing side|Day 3: Worksheet C |
| | |lengths | |
|4 | |13.2 Quadrants & Angles, |p. 939 # 1-9, 18-25, 42-49 |
| | |Coterminal angles, | |
| | |and terminal side points | |
|5 | |13.2-3 Reference Angles and Radian Measure |p. 939 # 10-17 |
| | | |p. 947 # 1-9, 19-26 |
|6 | |13.3 The Unit Circle |P. 947 # 10-17, 27-34, 36-38, 47, 49 |
|7 | |13.3 The Unit Circle Part 2 |Unit Circle Practice! |
| | |& Intro to Trig Inverses using the Unit Circle | |
| | | |Make certain you can fill one out completely &|
| | | |accurately! |
|8 | |Short unit circle quiz then: |p. 953 # 5-11, |
| | |13.4 Inverse Trig Functions |12-15 (ignore restrictions) |
| | | |19-24, 30 |
|9 | |REVIEW |Packet Problems (taken from |
| | | |Pages 976-978: Study Guide for Sections 13.1 |
| | | |to 13.4) |
|10 | | THE LAST TEST! | |
Trigonometry comes from the Greek words meaning _______________ _____________________.
There are ______ trig functions, and we are going to learn ______ of them. _________________!
A trig function is basically a ratio (fraction) of 2 sides of a right triangle…
➢ The three sides of the triangle are labeled as “adjacent,” “opposite,” and “hypotenuse.”
Examples: Label each of the 3 sides with adj, opp, or hyp.
|Main Trigonometric Functions |Reciprocal Trigonometric Functions |
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|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |[pic] |
Example: Find the value of the six trig functions.
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|[pic] |[pic] |
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|[pic] |[pic] |
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|[pic] |[pic] |
Sometimes you will need to find a missing side of the triangle…
* Use __________________ _________________!
Missing side: __________________
|Stations Activity |
|Station #1 |Station #2 |Station #3 |Station #4 |
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| |[pic] |[pic] |[pic] |
|[pic] | | | |
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| |[pic] |[pic] |[pic] |
|[pic] | | | |
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| |[pic] |[pic] |[pic] |
|[pic] | | | |
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| |[pic] |[pic] |[pic] |
|[pic] | | | |
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| |[pic] |[pic] |[pic] |
|[pic] | | | |
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| |[pic] |[pic] |[pic] |
|[pic] | | | |
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[pic]
[pic]
Write all fractions in simplest form.
[pic]
[pic] [pic] [pic]
Write the trigonometric function that is the reciprocal of each of the functions listed below.
4. [pic]
5. [pic]
6. [pic]
[pic]
The missing side length is:
_______________ _____________ _____________
[pic] [pic] [pic]
You have learned about special right triangles (45 – 45 – 90 & 30 – 60 – 90) before. Today we are going to evaluate trig functions for those special angles…
[pic]
If you know the ratios of the special right triangles, and you know the definitions of the trig functions, you can evaluate these angles easily. Let’s review the definitions of the trig functions…
|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |
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Now that we know the trig ratios and their values for 30°, 45°, and 60° we can use them to evaluate special right triangles with different side lengths. Take a look…
[pic]
Try These…
[pic]
[pic]
Extension: HW Start!
[pic]
[pic]
[pic]
[pic]
[pic]
Use trig functions and what you know about special right triangles to find x, the missing side, in each right triangle below.
[pic]
[pic] x = ___________
[pic] x = ___________
[pic] x = ___________
[pic] x = ___________
[pic] x = ___________
Quick review: Find the values of the 6 trig functions for the triangle below…
Now we are going to use trig ratios to solve real world problems – because that’s what math is all about!
1. A builder is constructing a wheelchair ramp from the ground to a deck whose height is 24 inches. The angle between the ground and the ramp is 5 degrees. To the nearest inch, what should the distance be from the end of the ramp to the deck, along the ground?
- draw a picture
- What side do you have to find?
- What trig function can you use?
- Write your equation with the trig function and a ratio of 2 sides.
- Solve the equation. Answer the question.
2. In a waterskiing competition, a jump ramp is 19 feet long and the measure of the angle between the ramp and the water is 15.1 degrees. To the nearest foot, what is the height of the top of the ramp above the water?
- draw a picture
- What side do you have to find?
- What trig function can you use?
- Write your equation with the trig function and a ratio of 2 sides.
- Solve the equation. Answer the question.
Angle of Elevation:
Angle of Depression:
3. A biologist whose eye level is 6 feet above the ground measures the angle of elevation to the top of a tree to be 38.7 degrees. If the biologist is standing 180 feet from the tree, what is the height of the tree to the nearest foot?
- draw a picture
- What side do you have to find?
- What trig function can you use?
- Write your equation with the trig function and a ratio of 2 sides.
- Solve the equation. Answer the question.
4. The pilot of a hot-air balloon measures the angle of depression to a landing spot to be 20.5 degrees. If the balloon’s altitude is 90 meters, what is the horizontal distance between the balloon and the landing spot?
- draw a picture
- What side do you have to find?
- What trig function can you use?
- Write your equation with the trig function and a ratio of 2 sides.
- Solve the equation. Answer the question.
[pic]
[pic]
[pic]
[pic]
[pic]
1c) [pic]
2c) [pic]
3c) [pic]
4c) [pic]
Warmup: Name the Quadrants and axes Today we will talk about angles of ______________.
|Drawing Rotational Angles |
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Practice: Draw the following rotational angles and state the quadrant of axis the terminal side is in…
1) 60° 2) 300° 3) 135° 4) 200°
|Positive vs. Negative Angles |
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|Rotation is _____________________ |Rotation is _________________________ |
Practice: Draw the following rotational angles and state the quadrant of axis the terminal side is in…
1) -20° 2) 120° 3) -240° 4) -380°
Angles that have the same terminal (ending side) are said to be ____________________ angles.
Example:
What do you think a coterminal angle with 60° would be?
|Finding Coterminal Angles! |
| |
|Simply _______ or subtract _____________ to the angle. (tomorrow we will look at radians!) |
Directions: Find a coterminal angle between -360° and 360°.
1) -40° 2) 300° 3) -100° 4) 600°
Now, sketch both angles on the same coordinate planes below…
Quick Refresher: The six trig functions…
|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |
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An angle can also be given by stating a point (in the coordinate plane) that lies on the terminal side
Example #1: [pic] Example #2: [pic]
Now, draw the angle and label it as [pic], and evaluate the 6 trig functions for [pic].
Practice on your own…
[pic]
Warmup: Evaluate each of the following (Hint: draw the special right triangles & use the trig definitions)
1) [pic] 2) [pic] 3) [pic]
We are going to learn to evaluate trig functions for rotational angles later in this unit…
To do this, we must understand the idea of a ____________________ angle.
Reference angles allow us to use the special right triangles (like the warmup) to evaluate rotational angles.
Definition: The reference angle is the positive _____________ angle formed by the terminal side of the
angle and the ____________________ x-axis.
| |[pic] |
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|Examples: | |
There are two methods to find a reference angles…
Method 1: Draw a picture.
Method 2: Use the formula (one for each quadrant)
|Finding Reference Angles |
|Quadrant I |Quadrant II |Quadrant III |Quadrant IV |
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|Ref [pic] = |Ref [pic] = |Ref [pic] = |Ref [pic] = |
Practice:
[pic]
[pic]
So far we have measured all angles in degrees. However, there are other units of measure…
[pic] [pic]
|Understanding Radians as a Fraction of 2π |
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|Degrees |
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|Degrees ( Radians |Example: |Radians ( Degrees |Example: |
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| |150° | |[pic] |
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More examples together…
1) [pic] 2) [pic] 3) [pic]
Practice on your own:
[pic]
Warmup: Find missing sides:
Trig values of rotational angles can be evaluated by using the unit circle.
A unit circle is a circle with __________________________.
*Powerpoint Presentation*
Each of the coordinates on the unit circle can give us the sin and cos of the individual angles.
[pic]
Let’s try a few (look at the unit circle).
1) [pic] 2) [pic] 3) [pic]
Because the tan function, and the 3 reciprocal functions are based upon the sin and cos, we can evaluate all 6 functions using the unit circle…
|Trig Functions on the Unit Circle: (x, y) = (cos, sin) |
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|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |
[pic]
If you have an entire unit circle, then that is probably the easiest way to evaluate these functions.
However, sometimes you don’t have a full unit circle… use _______________ _______________.
For each function below, first determine the quadrant of the terminal side, then find the reference angle.
1) [pic] 2) [pic] 3) [pic]
Quad:
Ref [pic]:
Value:
By using the reference angles, you really only need to memorize the coordinates of three angles…
30° = ________ ; 45° = ________ ; 60° = ________
[pic] [pic] [pic]
Try the following using reference angles…
[pic]
Unit Circle Practice!
Practice for tomorrow’s mini quiz:
Fill in the unit circle,
then evaluate the functions below!
1) [pic]
2) [pic]
3) [pic]
4) [pic]
Once you finish your practice quiz…
Take a look at one of the many unit circles you have
completed to answer the following questions…
Basically you are going to work backwards…. The first one is done for you…
Note: Give all answers in degrees (for now). There should be two answers for each!
1) [pic]
Answer: 45° or 135°
2) [pic] 3) [pic] 4) [pic]
5) [pic] 6) [pic] 7) [pic]
8) [pic] 9) [pic] 10) [pic]
“Working backwards” like this is known as an ____________________ function.
Rather than writing [pic], we will write _________________________.
Directions: Find the following (in both degree and radian)
1) [pic]° 2) [pic]° 3) [pic]°
= ________ radians = ________ radians = ________ radians
*** Important Note *** for [pic] and [pic] the numbers must be between ______ and _______ !!!!
Directions: Fill in both unit circles completely.
Make sure you are able to do this to start class tomorrow!
Important Note about Trig Functions vs. Trig Inverses:
[pic]
(applies to all the trig functions)
[pic]
Yesterday we learned how to evaluate [pic], but what about problems like[pic]
Graphing Calculator Time!
MAKE SURE THAT YOUR CALCULATOR IS IN THE RIGHT MODE!!!
Degrees will give you answers between __________ and ___________.
Radians will give you answers between __________ and ___________.
Remember, for sin and cos, you must enter numbers between ______ and _______!
Directions: Evaluate each trig function below. Give your answer in degree & radian.
1) [pic] D: _______ R: ________ 2) [pic] D: _______ R: ________
3) [pic] D: _______ R: ________ 4) [pic] D: _______ R: ________
5) [pic] D: _______ R: ________ 6) [pic] D: _______ R: ________
7) [pic] D: _______ R: ________ 8) [pic] D: _______ R: ________
Shift Gears: Back to the unit circle.
1) [pic]° 2) [pic]° 3) [pic]°
= ________ radians = ________ radians = ________ radians
4) [pic]° 5) [pic]° 6) [pic]°
= ________ radians = ________ radians = ________ radians
Additional Practice (if needed)
[pic]
[pic]
[pic]
[pic]
[pic]
This is your last test, make it a good one!
-----------------------
Date _________ Period_________
Unit 11: Trigonometry
& Equation
Day 1: Trigonometric Functions Introduction
& Equation
[pic]
[pic]
[pic]
[pic]
5
4
3
Now find the 6 trig functions:
[pic] [pic]
[pic] [pic]
[pic] [pic]
13
[pic]
12
HOMEWORK DAY 1: WORKSHEET A
& Equation
Day 2: Special Right Triangles
& Equation
1
1
[pic]
[pic]
[pic]
HOMEWORK DAY 2: WORKSHEET B
& Equation
Day 3: Solving Right Triangles – Finding Missing Side Lengths
& Equation
7
25
HOMEWORK DAY 3: WORKSHEET C
& Equation
Day 4: Quadrants, Coterminal Angles, and Terminal Side Points
& Equation
-270°
90°
Day 5: Reference Angles and Radian Measure
& Equation
Day 6: The Unit Circle
& Equation
30°
60°
1
45°
45°
1
Signs of Quadrants
Here’s the Answer…
What is the Question?!
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Day 7 Homework: Unit Circle Practice, Practice!
& Equation
Day 8: Inverse Trig Functions from the Unit Circle
& Equation
Day 9: Trig Review
& Equation
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