Materials Needed

• Scientific calculator (physical or online)
• Ruler (metric)
• Protractor
• Graph paper and pencils
• Colored highlighters (3 colors)
• A small object to measure (e.g., a phone, a book, or a cereal box)

Learning Objectives

By the end of this lesson, you will be able to:

• Correctly identify the Hypotenuse, Opposite, and Adjacent sides of a right-angled triangle relative to a given angle.
• Define and calculate the three primary trigonometric ratios: Sine (sin), Cosine (cos), and Tangent (tan).
• Use the SOH CAH TOA mnemonic to solve ratio problems.

1. Introduction: The "How High?" Hook (8 Minutes)

The Scenario: Imagine you are a drone pilot. You need to fly your drone to the top of a 50-foot statue to inspect it. You are standing 30 feet away from the base. At what angle do you need to tilt your camera to see the top? Or, if you know the angle, how do you calculate the distance the drone has to fly?

Trigonometry is essentially the "GPS" of mathematics. It’s the study of how angles relate to side lengths. Today, we are cracking the code that video game developers, architects, and NASA engineers use every day.

Success Criteria: You’ll know you’ve nailed this lesson if you can look at any right triangle and immediately write down its three magic ratios (Sin, Cos, and Tan) without checking your notes.

2. I Do: Anatomy of the Triangle (10 Minutes)

Before we calculate, we have to label. Everything in Trig depends on your Reference Angle (usually called Theta or θ).

• Hypotenuse: The longest side. Always across from the 90-degree right angle. It never changes.
• Opposite: The side directly across from your reference angle θ. Think of it as the side the angle "looks at."
• Adjacent: The side "next to" your reference angle that isn't the hypotenuse.

The Legend of SOH CAH TOA:

• SOH: Sin(θ) = Opposite / Hypotenuse
• CAH: Cos(θ) = Adjacent / Hypotenuse
• TOA: Tan(θ) = Opposite / Adjacent

Example: If a triangle has an Opposite side of 3cm and a Hypotenuse of 5cm, the Sine ratio is 3/5 or 0.6.

3. We Do: The Triangle Construction (15 Minutes)

Let’s build one together to see if the math actually works in the real world.

1. On your graph paper, use a ruler to draw a horizontal line 8cm long (this is your Adjacent side).
2. Use your protractor to draw a 90-degree angle at one end. Draw a vertical line 6cm high (this is your Opposite side).
3. Connect the ends to form the Hypotenuse.
4. Measure: Use your ruler to measure the hypotenuse. It should be exactly 10cm.
5. The Check: Let's calculate the ratios for the angle where you started:
   • Sin = 6/10 (0.6)
   • Cos = 8/10 (0.8)
   • Tan = 6/8 (0.75)
6. Calculator Verification: Use your protractor to measure the angle. It should be about 36.9 degrees. Type "sin(36.9)" into your calculator. Does it come close to 0.6? (Note: Ensure calculator is in DEGREE mode).

4. You Do: The "Ramp Design" Challenge (15 Minutes)

Task: You are designing a skate ramp or a wheelchair ramp. Safety regulations say the ramp must have a specific "steepness" (ratio).

Pick a small object near you (like a book). Lean it against a wall to create a triangle.

1. Measure the length of the object (the Hypotenuse).
2. Measure how far the base is from the wall (the Adjacent side).
3. Use the Pythagorean theorem (a² + b² = c²) or measure the height to find the Opposite side.
4. Calculate the Sin, Cos, and Tan for the angle the object makes with the floor.
5. Write your results as both a fraction and a decimal.

Challenge: If you changed the angle to be steeper, which ratio (Sin, Cos, or Tan) would increase the most? Write a one-sentence prediction.

5. Adaptations & Extensions

• If you're struggling: Focus only on labeling the sides. Use three different colored highlighters: always highlight the Hypotenuse in yellow, the Opposite in orange, and the Adjacent in pink. Practice this on 5 different-sized triangles before doing the math.
• If you want a challenge: Explore the "Inverse" functions on your calculator (sin⁻¹, cos⁻¹, tan⁻¹). If you know the ratio is 0.6, can you use the inverse button to find the exact angle without a protractor?

6. Conclusion & Recap (5 Minutes)

The Summary: Trigonometry isn't about scary formulas; it’s about ratios. No matter how big or small a triangle is, if the angles are the same, the ratio of the sides will always be the same. SOH CAH TOA is your key to unlocking those ratios.

Quick Fire Check:

• Which side is always across from the right angle?
• If I have the "Opposite" and the "Adjacent," which trig function should I use?
• What does the "O" and "H" stand for in SOH?

Preview for Lesson 2: Next time, we will use these ratios to find missing side lengths—basically, using math to measure things that are too tall to reach with a ruler!

Assessment

Formative: Observation of the "Ramp Design" calculations and the calculator verification step.

Summative: Create a "Trig Cheat Sheet" card. On one side, draw a right triangle with labeled sides. On the other, write the SOH CAH TOA definitions. Use this to solve three practice triangles provided by your instructor or textbook.