Lesson Plan
Unit Circle & Trig Unlocked!
Students will be able to define and apply sine, cosine, and tangent using the unit circle and understand the relationship between angles and trigonometric values.
Trigonometry is a fundamental concept in advanced mathematics and science, essential for understanding waves, oscillations, and more. Mastering the unit circle provides a powerful tool for solving complex problems and visualizing these functions.
Audience
12th Grade
Time
45 minutes
Approach
Interactive lecture, guided practice, and collaborative problem-solving.
Prep
Teacher Preparation
15 minutes
- Review the Unit Circle & Trig Unlocked! Slide Deck and familiarize yourself with the content.
- Print copies of the Unit Circle Practice Worksheet for each student.
- Prepare to project the slide deck.
- Review the Unit Circle Practice Answer Key for grading and discussion.
- Ensure all technology (projector, computer) is working correctly.
Step 1
Warm-Up: What's Your Angle?
5 minutes
- Begin with the Trig Warm-Up activity.
- Project the warm-up question on the board.
- Ask students to briefly jot down their thoughts or discuss with a partner their initial ideas about angles and circles. (e.g., "What do you already know about angles and circles? How could a point on a circle describe an angle?")
- Briefly discuss student responses, connecting them to prior knowledge.
Step 2
Introduction: The Unit Circle Unveiled (Slides 1-4)
10 minutes
- Use Unit Circle & Trig Unlocked! Slide Deck slides 1-4.
- Introduce the concept of the unit circle: a circle with radius 1 centered at the origin.
- Explain how points on the unit circle relate to angles in standard position.
- Define sine, cosine, and tangent in terms of x, y, and r (where r=1 for the unit circle).
- Emphasize that cosine is the x-coordinate and sine is the y-coordinate on the unit circle.
Step 3
Exploring Key Angles (Slides 5-8)
10 minutes
- Use Unit Circle & Trig Unlocked! Slide Deck slides 5-8.
- Guide students through identifying coordinates for key angles (0°, 90°, 180°, 270°, 360° or 0, π/2, π, 3π/2, 2π radians).
- Discuss special right triangles (30-60-90 and 45-45-90) to derive coordinates for 30°, 45°, 60° and their radian equivalents.
- Encourage students to find patterns and symmetries within the unit circle.
Step 4
Practice & Application: Worksheet Time
15 minutes
- Distribute the Unit Circle Practice Worksheet.
- Instruct students to work individually or in pairs to complete the worksheet, applying what they've learned about the unit circle and trig functions.
- Circulate around the room to provide support and answer questions.
- If time permits, review some answers using the Unit Circle Practice Answer Key as a class.
Step 5
Cool-Down: Reflect and Connect
5 minutes
- Conclude the lesson with the Trig Cool-Down activity.
- Project the cool-down question.
- Ask students to write a brief reflection on what they learned or one concept they found challenging/interesting.
- Collect cool-down responses as an exit ticket.
Slide Deck
Unit Circle & Trig Unlocked!
Demystifying Trigonometry, One Circle at a Time!
Welcome students and introduce the exciting journey into understanding the unit circle and trigonometric functions. Emphasize that this lesson will simplify these concepts.
Warm-Up: What's Your Angle?
What do you already know about angles and circles?
How could a point on a circle describe an angle?
Display the warm-up question. Give students a minute or two to think individually, then ask them to discuss with a partner. Encourage sharing of initial thoughts and connections to prior knowledge. This primes their thinking for the unit circle concept.
Meet the Unit Circle
- A circle with a radius of 1 unit.
- Centered at the origin (0,0) of a coordinate plane.
- It's our secret weapon for understanding trigonometry!
Introduce the unit circle. Explain that it's a special circle that makes understanding trigonometry much easier. Emphasize its center at the origin (0,0) and a radius of 1. Use a visual to show this.
Angles in Standard Position
- An angle starts at the positive x-axis (initial side).
- Rotates counter-clockwise to its terminal side.
- The point where the terminal side meets the unit circle is (x, y).
Explain how an angle in standard position interacts with the unit circle. Define the initial and terminal sides. The key takeaway is that the point where the terminal side intersects the unit circle is crucial.
Trig Functions on the Unit Circle
For any point (x, y) on the unit circle corresponding to an angle θ:
- Cosine (cos θ) = x-coordinate
- Sine (sin θ) = y-coordinate
- Tangent (tan θ) = y/x (where x ≠ 0)
Now, introduce the definitions of sine, cosine, and tangent based on the unit circle. Stress the direct relationship: cosine is x, sine is y. Briefly explain tangent as y/x. You can mention 'SOH CAH TOA' as a memory aid but emphasize the unit circle connection here.
Key Angles: The Quadrantal Crew
Let's find the (x, y) coordinates and the sin, cos, tan values for these angles:
- 0° (0 radians)
- 90° (π/2 radians)
- 180° (π radians)
- 270° (3π/2 radians)
- 360° (2π radians)
Start with the cardinal angles. Ask students to identify the coordinates for 0°, 90°, 180°, and 270°. Relate these to the axes. This builds confidence before moving to more complex angles.
Special Triangles: 30°-60°-90°
Remember our special right triangles?
- Use the 30°-60°-90° triangle to find the coordinates for:
- 30° (π/6 radians)
- 60° (π/3 radians)
Introduce the 30-60-90 special right triangle. Explain how it fits into the unit circle to find the coordinates for 30° and 60°. Visual aids of the triangle within the circle are very helpful here.
Special Triangles: 45°-45°-90°
- Use the 45°-45°-90° triangle to find the coordinates for:
- 45° (π/4 radians)
Similarly, cover the 45-45-90 special right triangle. Show how it helps find the coordinates for 45°. Emphasize the symmetry and how once you know the first quadrant, you can find values in others.
Time to Practice!
Now it's your turn to apply what you've learned!
Work on the Unit Circle Practice Worksheet.
Don't be afraid to collaborate with a partner!
Transition to the worksheet. Explain that practicing will solidify their understanding. Encourage them to refer back to the unit circle concepts and special triangles. Offer guidance as they work.
Cool-Down: Reflect and Connect
In your own words, describe one new thing you learned about the unit circle or trig functions today. Or, what was one concept that you found most interesting/challenging?
For the cool-down, ask students to reflect on their learning. This helps them consolidate information and provides you with feedback on areas of understanding or difficulty.
Warm Up
Trig Warm-Up: What's Your Angle?
Instructions: Take a few moments to think about the questions below. You can jot down your answers or discuss them with a partner.
1. What do you already know about angles and circles? Think about degrees, rotation, or how we measure them.
2. Imagine a point moving around a circle. How could the position of this point describe an angle? What kind of information would you need?
Worksheet
Unit Circle Practice Worksheet
Instructions: Use your knowledge of the unit circle and trigonometric functions to complete the following exercises. Show your work where applicable.
## Part 1: Coordinates on the Unit Circle
For each angle, identify the (x, y) coordinates of the point on the unit circle.
1. Angle: 0° (0 radians)
Coordinates:
2. Angle: 90° (π/2 radians)
Coordinates:
3. Angle: 180° (π radians)
Coordinates:
4. Angle: 270° (3π/2 radians)
Coordinates:
5. Angle: 30° (π/6 radians)
Coordinates:
6. Angle: 45° (π/4 radians)
Coordinates:
7. Angle: 60° (π/3 radians)
Coordinates:
## Part 2: Trigonometric Values
For each angle, find the exact values of sin θ, cos θ, and tan θ.
8. Angle: 0°
sin(0°) =
cos(0°) =
tan(0°) =
9. Angle: 90°
sin(90°) =
cos(90°) =
tan(90°) =
10. Angle: 180°
sin(180°) =
cos(180°) =
tan(180°) =
11. Angle: 270°
sin(270°) =
cos(270°) =
tan(270°) =
12. Angle: 30°
sin(30°) =
cos(30°) =
tan(30°) =
13. Angle: 45°
sin(45°) =
cos(45°) =
tan(45°) =
14. Angle: 60°
sin(60°) =
cos(60°) =
tan(60°) =
Answer Key
Unit Circle Practice Answer Key
Part 1: Coordinates on the Unit Circle
For each angle, identify the (x, y) coordinates of the point on the unit circle.
-
Angle: 0° (0 radians)
- Thought Process: At 0 degrees, the terminal side lies along the positive x-axis. Since the radius is 1, the point on the unit circle is 1 unit away from the origin along the positive x-axis.
- Coordinates: (1, 0)
-
Angle: 90° (π/2 radians)
- Thought Process: At 90 degrees, the terminal side lies along the positive y-axis. The point on the unit circle is 1 unit away from the origin along the positive y-axis.
- Coordinates: (0, 1)
-
Angle: 180° (π radians)
- Thought Process: At 180 degrees, the terminal side lies along the negative x-axis. The point on the unit circle is 1 unit away from the origin along the negative x-axis.
- Coordinates: (-1, 0)
-
Angle: 270° (3π/2 radians)
- Thought Process: At 270 degrees, the terminal side lies along the negative y-axis. The point on the unit circle is 1 unit away from the origin along the negative y-axis.
- Coordinates: (0, -1)
-
Angle: 30° (π/6 radians)
- Thought Process: For a 30-60-90 triangle with hypotenuse 1 (the radius), the side opposite 30° is 1/2, and the side opposite 60° is √3/2. In the first quadrant, x is the adjacent side (to the x-axis) and y is the opposite side.
- Coordinates: (√3/2, 1/2)
-
Angle: 45° (π/4 radians)
- Thought Process: For a 45-45-90 triangle with hypotenuse 1, both legs are √2/2. In the first quadrant, x and y are both positive.
- Coordinates: (√2/2, √2/2)
-
Angle: 60° (π/3 radians)
- Thought Process: For a 30-60-90 triangle with hypotenuse 1, the side opposite 60° is √3/2, and the side opposite 30° is 1/2. In the first quadrant, x is the adjacent side (to the x-axis) and y is the opposite side.
- Coordinates: (1/2, √3/2)
Part 2: Trigonometric Values
For each angle, find the exact values of sin θ, cos θ, and tan θ. (Remember: cos θ = x, sin θ = y, tan θ = y/x)
-
Angle: 0° (Coordinates: (1, 0))
- sin(0°) = y = 0
- cos(0°) = x = 1
- tan(0°) = y/x = 0/1 = 0
-
Angle: 90° (Coordinates: (0, 1))
- sin(90°) = y = 1
- cos(90°) = x = 0
- tan(90°) = y/x = 1/0 = Undefined
-
Angle: 180° (Coordinates: (-1, 0))
- sin(180°) = y = 0
- cos(180°) = x = -1
- tan(180°) = y/x = 0/-1 = 0
-
Angle: 270° (Coordinates: (0, -1))
- sin(270°) = y = -1
- cos(270°) = x = 0
- tan(270°) = y/x = -1/0 = Undefined
-
Angle: 30° (Coordinates: (√3/2, 1/2))
- sin(30°) = y = 1/2
- cos(30°) = x = √3/2
- tan(30°) = y/x = (1/2) / (√3/2) = 1/√3 = √3/3
-
Angle: 45° (Coordinates: (√2/2, √2/2))
- sin(45°) = y = √2/2
- cos(45°) = x = √2/2
- tan(45°) = y/x = (√2/2) / (√2/2) = 1
-
Angle: 60° (Coordinates: (1/2, √3/2))
- sin(60°) = y = √3/2
- cos(60°) = x = 1/2
- tan(60°) = y/x = (√3/2) / (1/2) = √3
Cool Down
Trig Cool-Down: Reflect and Connect
Instructions: Take a few minutes to respond to one of the prompts below. Your reflection will help me understand what you learned today!
1. In your own words, describe one new thing you learned about the unit circle or trigonometric functions today.
2. What was one concept that you found most interesting or most challenging about today's lesson? Why?