Trig Warm-Up: What's Your Angle?

Directions: Take a few minutes to answer the following questions to get your brain ready for trigonometry!

1. Draw a right-angled triangle. Label the right angle.





2. What is the sum of the angles in any triangle?



3. If one acute angle in a right triangle is 40 degrees, what is the measure of the other acute angle?



4. In a right triangle, what is the longest side called? Which angle is it always opposite?



5. Imagine you are standing at one of the acute angles of your right triangle. How would you describe the side directly across from you?

Trig Script: Right Triangle Ratios

Warm-Up: What's Your Angle? (5 minutes)

Teacher: "Good morning, everyone! Let's get our brains warmed up for some exciting math. Please take out the Trig Warm-Up I've just handed out. You have about 5 minutes to complete it."

Allow students to work individually. Circulate to check for understanding and address initial questions.

Teacher: "Alright, let's review these together. For number 1, who can describe their right triangle?"
Call on a student. Discuss and affirm key features.
Teacher: "Excellent. And number 2, what's the sum of angles in any triangle?"
Expect 180 degrees.
Teacher: "Correct! So, for a right triangle, if one acute angle is 40 degrees, what's the other acute angle?"
Guide students to subtract 90 and 40 from 180, getting 50 degrees.
Teacher: "Fantastic! And the longest side of a right triangle? What's it called and which angle is it opposite?"
Expect hypotenuse, opposite the right angle.
Teacher: "Perfect! Lastly, if you're at an acute angle, how would you describe the side directly across from you?"
Expect 'across from it' or 'on the other side'. This sets up 'opposite'.

Introduction to Trigonometric Ratios (15 minutes)

Teacher: "Today, we're going to dive into a super useful branch of mathematics called Trigonometry! Don't let the big word scare you. It simply means 'triangle measurement.'"
(Display Right Triangle Ratios Slide Deck - Slide 3)
Teacher: "Trigonometry is all about the special relationships between the angles and sides of right triangles. We use it to find missing side lengths or angles when we have some information, which is super handy in fields like engineering, architecture, and even video game design!"

(Display Right Triangle Ratios Slide Deck - Slide 4)
Teacher: "To help us remember these relationships, we have a fun little acronym: SOH CAH TOA. Say it with me: SOH CAH TOA!"
Have students repeat.
Teacher: "This mnemonic helps us remember the three basic trigonometric ratios: Sine, Cosine, and Tangent. Each one is a fraction, a ratio, that compares two sides of a right triangle relative to a specific acute angle."

(Display Right Triangle Ratios Slide Deck - Slide 5)
Teacher: "Before we get to the ratios, it's crucial we know how to label the sides of a right triangle from the perspective of one of the acute angles. Let's say we're focusing on Angle A. Who can remind me what the longest side, opposite the right angle, is called?"
Expect 'hypotenuse'.
Teacher: "That's right, the Hypotenuse. This side never changes. But the other two sides change depending on which acute angle we are looking from."

"If we are looking from Angle A:
* The side directly Opposite Angle A is called the Opposite side.
* The side that is next to Angle A, but not the hypotenuse, is called the Adjacent side. 'Adjacent' just means 'next to'."

Draw a right triangle on the board. Label vertices A, B, C with C as the right angle. Pick Angle A. Ask students to identify opposite, adjacent, hypotenuse. Then pick Angle B and repeat.

Teacher: "Any questions on identifying these sides? It's really important to get this right!"

(Display Right Triangle Ratios Slide Deck - Slide 6)
Teacher: "Now, let's break down SOH CAH TOA. Each part tells us how to form a ratio:
* SOH stands for Sine = Opposite / Hypotenuse
* CAH stands for Cosine = Adjacent / Hypotenuse
* TOA stands for Tangent = Opposite / Adjacent

"You'll use your calculator to find the sine, cosine, or tangent of an angle. Make sure your calculator is in degree mode!"

Guided Practice: Finding Missing Sides (20 minutes)

Teacher: "Let's put this into practice and find some missing side lengths!"

(Display Right Triangle Ratios Slide Deck - Slide 7)
Teacher: "Here's our first example. We have a right triangle with a 30-degree angle. We know the hypotenuse is 10, and we want to find the side opposite the 30-degree angle, which we've labeled 'x'."
"First step: Identify your angle of focus – here it's 30 degrees. Second step: Label the sides you know and the side you want to find, relative to that angle. We have the Opposite (x) and the Hypotenuse (10)."
"Which of our SOH CAH TOA ratios uses Opposite and Hypotenuse?"
Wait for responses. Guide towards SOH.
Teacher: "That's SOH! So we'll use Sine. Our equation will be: sin(30°) = x / 10. To solve for x, what should we do?"
Guide students to multiply both sides by 10.
Teacher: "So, x = 10 * sin(30°). Grab your calculators! What do you get for x?"
Expect x = 5.
Teacher: "Great job! Does an answer of 5 make sense in this triangle? Why or why not?"
Discuss the reasonableness of the answer.

(Display Right Triangle Ratios Slide Deck - Slide 8)
Teacher: "Next example! We have a 45-degree angle. We want to find the adjacent side, labeled 'y', and we know the hypotenuse is 7. What's our first step?"
Students identify angle 45 degrees.
Teacher: "Good. And the sides? We have Adjacent (y) and Hypotenuse (7). Which ratio fits?"
Guide towards CAH.
Teacher: "CAH it is, so we use Cosine. Set up the equation: cos(45°) = y / 7. How do we solve for y?"
Guide students to multiply both sides by 7.
Teacher: "Calculators out! What is y = 7 * cos(45°)?"
Expect y ≈ 4.95.
Teacher: "Excellent! Notice that we often get decimal answers here. That's perfectly normal for trigonometry."

(Display Right Triangle Ratios Slide Deck - Slide 9)
Teacher: "Last guided example for finding a side! We have a 60-degree angle. We know the adjacent side is 8, and we want to find the opposite side, 'z'. What are our known sides relative to the 60-degree angle?"
Expect Opposite and Adjacent.
Teacher: "Which SOH CAH TOA ratio uses Opposite and Adjacent?"
Guide towards TOA.
Teacher: "TOA means we use Tangent. Our equation: tan(60°) = z / 8. Solve for z!"
Guide students to multiply both sides by 8.
Teacher: "What do you get for z = 8 * tan(60°)?"
Expect z ≈ 13.86.
Teacher: "Fantastic! Does anyone have any questions on these examples before you try some on your own?"

Independent Practice: SohCahToa Worksheet (15 minutes)

(Display Right Triangle Ratios Slide Deck - Slide 10)
Teacher: "Alright, it's your turn to practice! I'm handing out the SohCahToa Practice Worksheet. Your task is to use SOH CAH TOA to find the missing side lengths in each right triangle. Remember to show your work and use your calculators carefully. I will be circulating to help if you get stuck."

Distribute the worksheets. Circulate and provide individual support as needed. Remind students to label their triangles and choose the correct ratio. Provide encouragement.

Teacher: "We'll wrap up the worksheet in about 15 minutes. If you don't finish, it will be homework."

Cool-Down: Trig Ticket Out (5 minutes)

Teacher: "Okay everyone, eyes up here for just a moment. To finish our lesson today, I'm handing out a quick Trig Cool-Down. This is your 'ticket out the door' today. Please answer the question to the best of your ability. It helps me see what you understood today."

Distribute the cool-down. Collect them as students complete them. Ensure all students submit a cool-down.

Teacher: "Great work today, mathematicians! Remember SOH CAH TOA! We'll continue with trigonometry next time."

SohCahToa Practice: Finding Missing Sides

Directions: For each right-angled triangle, use the given information and the SOH CAH TOA mnemonic to find the length of the missing side. Round your answers to two decimal places.

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## Problem 1
In a right triangle, an angle measures 35 degrees. The hypotenuse is 15 units long. Find the length of the side opposite the 35-degree angle.







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## Problem 2
In a right triangle, an angle measures 50 degrees. The side adjacent to the 50-degree angle is 8 units long. Find the length of the hypotenuse.







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## Problem 3
In a right triangle, an angle measures 62 degrees. The side opposite the 62-degree angle is 12 units long. Find the length of the side adjacent to the 62-degree angle.







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## Problem 4
A ladder leans against a wall, forming a 70-degree angle with the ground. The base of the ladder is 4 feet away from the wall. How long is the ladder?







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## Problem 5
A kite is flying at an angle of elevation of 42 degrees. The string connecting the kite to the ground is 100 meters long. How high is the kite above the ground?

SohCahToa Practice: Answer Key

Directions: Here are the step-by-step solutions for the SohCahToa Practice Worksheet.

Problem 1

In a right triangle, an angle measures 35 degrees. The hypotenuse is 15 units long. Find the length of the side opposite the 35-degree angle.

Thought Process:

1. Identify the angle: 35 degrees.
2. Label the knowns and unknowns relative to the angle:
   • Opposite = x (unknown)
   • Hypotenuse = 15 (known)
3. Choose the correct trigonometric ratio: We have Opposite and Hypotenuse, so we use SOH (Sine = Opposite / Hypotenuse).
4. Set up the equation: sin(35°) = x / 15
5. Solve for x:
   x = 15 * sin(35°)
x ≈ 15 * 0.5736
x ≈ 8.60 units

Answer: The length of the side opposite the 35-degree angle is approximately 8.60 units.

Problem 2

In a right triangle, an angle measures 50 degrees. The side adjacent to the 50-degree angle is 8 units long. Find the length of the hypotenuse.

Thought Process:

1. Identify the angle: 50 degrees.
2. Label the knowns and unknowns relative to the angle:
   • Adjacent = 8 (known)
   • Hypotenuse = x (unknown)
3. Choose the correct trigonometric ratio: We have Adjacent and Hypotenuse, so we use CAH (Cosine = Adjacent / Hypotenuse).
4. Set up the equation: cos(50°) = 8 / x
5. Solve for x:
   x * cos(50°) = 8
x = 8 / cos(50°)
x ≈ 8 / 0.6428
x ≈ 12.45 units

Answer: The length of the hypotenuse is approximately 12.45 units.

Problem 3

In a right triangle, an angle measures 62 degrees. The side opposite the 62-degree angle is 12 units long. Find the length of the side adjacent to the 62-degree angle.

Thought Process:

1. Identify the angle: 62 degrees.
2. Label the knowns and unknowns relative to the angle:
   • Opposite = 12 (known)
   • Adjacent = x (unknown)
3. Choose the correct trigonometric ratio: We have Opposite and Adjacent, so we use TOA (Tangent = Opposite / Adjacent).
4. Set up the equation: tan(62°) = 12 / x
5. Solve for x:
   x * tan(62°) = 12
x = 12 / tan(62°)
x ≈ 12 / 1.8807
x ≈ 6.38 units

Answer: The length of the side adjacent to the 62-degree angle is approximately 6.38 units.

Problem 4

A ladder leans against a wall, forming a 70-degree angle with the ground. The base of the ladder is 4 feet away from the wall. How long is the ladder?

Thought Process:

1. Visualize/Draw the triangle: The ladder, the wall, and the ground form a right triangle. The angle with the ground is 70 degrees. The distance from the base of the ladder to the wall is the side adjacent to the 70-degree angle. The length of the ladder is the hypotenuse.
2. Identify the angle: 70 degrees.
3. Label the knowns and unknowns relative to the angle:
   • Adjacent = 4 feet (known)
   • Hypotenuse = x (length of the ladder, unknown)
4. Choose the correct trigonometric ratio: We have Adjacent and Hypotenuse, so we use CAH (Cosine = Adjacent / Hypotenuse).
5. Set up the equation: cos(70°) = 4 / x
6. Solve for x:
   x * cos(70°) = 4
x = 4 / cos(70°)
x ≈ 4 / 0.3420
x ≈ 11.70 feet

Answer: The ladder is approximately 11.70 feet long.

Problem 5

A kite is flying at an angle of elevation of 42 degrees. The string connecting the kite to the ground is 100 meters long. How high is the kite above the ground?

Thought Process:

1. Visualize/Draw the triangle: The kite, its string, and the ground form a right triangle. The angle of elevation is the angle between the ground and the string (42 degrees). The length of the string is the hypotenuse. The height of the kite is the side opposite the angle of elevation.
2. Identify the angle: 42 degrees.
3. Label the knowns and unknowns relative to the angle:
   • Opposite = x (height of the kite, unknown)
   • Hypotenuse = 100 meters (known)
4. Choose the correct trigonometric ratio: We have Opposite and Hypotenuse, so we use SOH (Sine = Opposite / Hypotenuse).
5. Set up the equation: sin(42°) = x / 100
6. Solve for x:
   x = 100 * sin(42°)
x ≈ 100 * 0.6691
x ≈ 66.91 meters

Answer: The kite is approximately 66.91 meters high above the ground.

Trig Cool-Down: Ticket Out the Door

Directions: Please answer the following question to the best of your ability.

Consider a right triangle with an angle of 25 degrees. If the side adjacent to this angle is 10 units long, explain which trigonometric ratio you would use to find the length of the hypotenuse, and why.