Right Triangle Trigonometry

Overview

Students learn to use right-triangle trigonometric ratios—sine, cosine, and tangent—to identify side relationships and calculate missing side lengths or angles. They solve short practical problems using correct opposite/adjacent/hypotenuse labeling.

Learning intentions

• Students will identify hypotenuse, opposite, and adjacent sides in a right triangle relative to a chosen angle.
• Students will use sine, cosine, and tangent ratios to calculate missing side lengths.
• Students will use trigonometric ratios to calculate missing angles in right triangles.
• Students will apply reasoning to choose the correct ratio and check whether answers make sense.

Success criteria

• I can correctly label hypotenuse, opposite, and adjacent for a given angle.
• I can write the correct sine, cosine, or tangent ratio and solve for the missing value.
• I can calculate a missing angle or side using trig ratios and round appropriately.
• I can explain why my chosen ratio matches the situation.

Curriculum links

• Applies right-triangle trigonometry to solve for missing sides and angles using sine, cosine, and tangent.
• Develops and applies mathematical reasoning, including justifying choices of strategies and interpreting solutions.
• Uses tools (technology/calculators) appropriately for trigonometric calculations and estimates for reasonableness.

Lesson structure (45 minutes)

1. (0–5) Warm-up: “Which side is which?” Display 3 right triangles (on board or slides). Ask students to point to hypotenuse first, then identify opposite and adjacent for a marked angle. Take quick cold-call answers and confirm as a class.

2. (5–12) Direct instruction: core ratio relationships Teach the rule: hypotenuse is always opposite the right angle; opposite/adjacent are relative to the chosen acute angle. Write the three ratios clearly:

• sin(angle) = opposite/hypotenuse
• cos(angle) = adjacent/hypotenuse
• tan(angle) = opposite/adjacent Emphasize that adjacent must be the non-hypotenuse side next to the angle.

3. (12–18) Guided example 1 (missing side) Provide one problem: a right triangle where an angle θ is given and one side is given (e.g., opposite and θ, then ask for hypotenuse). Step through:

• Label opposite/adjacent/hypotenuse relative to θ
• Choose the matching ratio (e.g., sin if opposite and hypotenuse are involved)
• Substitute values into the ratio
• Solve for the unknown side
• Use calculator and round (teacher models how to report units)

4. (18–26) Guided example 2 (missing angle) Provide a second problem: right triangle with two sides (e.g., adjacent and hypotenuse, ask for angle). Step through similarly, but include:

• Substitute into the correct ratio
• Use inverse trig to find the angle (sin⁻¹, cos⁻¹, or tan⁻¹)
• Check the result is an acute angle and consistent with the side relationship Briefly note that in a right triangle, θ is typically between 0° and 90° for the acute angles.

5. (26–37) Pair practice: solve and justify Students work in pairs on 4 short right-triangle questions of increasing complexity. Each question requires a labeled diagram and a written ratio (one sentence justification). Examples:

• Find a missing side using sine or cosine
• Find a missing angle using tangent or sine
• A word problem (practical context): for instance, ladder reach, ramp angle, or viewing angle Teacher circulates with a checklist: labeling is correct, ratio is selected correctly, calculator use is appropriate, and answers are plausible.

6. (37–43) Whole-class share: strategy and common errors Select two student pairs to share their work. Prompt class discussion on common mistakes: swapping adjacent/opposite, using the wrong inverse trig button, treating the hypotenuse as a leg, or mixing up which angle the ratio refers to.

7. (43–45) Exit ticket (quick check) Students complete one problem independently: identify sides for θ, then calculate a missing side or angle using the appropriate ratio. Collect for formative assessment.

Resources

• Printed or projected right-triangle practice worksheet (4 questions + exit ticket)
• Right-triangle reference diagram showing hypotenuse and angle-relative opposite/adjacent
• Calculators for each student (or shared devices)
• Teacher board markers or slide deck
• Angle measurement/rounding guidance (e.g., nearest tenth of a degree where appropriate)
• Worked example sheet (for students who need a model)
• Pair work roles card (Calculator/Explainer) to support reasoning

Assessment

• Formative checks during warm-up (correct side labeling) and guided practice (ratio selection).
• Teacher observation during pair work: students must label sides, write the ratio, and show substitution steps.
• Exit ticket to confirm students can both identify sides and apply sine/cosine/tangent to solve for a missing side/angle.

Differentiation

• Support: Provide a partially filled worked template for the first two practice questions (labeling prompts and sentence starters like “Because we know opposite and hypotenuse, I use sine.”).
• Support: Offer a “ratio matcher” card listing sin/cos/tan and what side pairs each one uses.
• Extension: Include a question where students must choose between sine and cosine based on given sides, then interpret the result in context (e.g., which direction the angle should change).
• EAL/SEN: Use consistent diagram language and sentence frames; allow verbal justification; provide extra time and reduced number of practice items if needed while keeping at least one “missing side” and one “missing angle” task.