Right Triangle Trig Ratios

Overview

Students learn to use sine, cosine, and tangent ratios to solve right triangles, finding missing side lengths and angle measures. They connect ratio reasoning to real triangle structures and check their work using inverse ratios and consistency.

Learning intentions

• Students will be able to identify the hypotenuse, adjacent, and opposite sides in a right triangle.
• Students will use sine, cosine, and tangent ratios to find missing side lengths.
• Students will use sine, cosine, and tangent ratios to find missing acute angles.
• Students will justify whether the chosen ratio matches the given angle and sides.

Success criteria

• I can correctly label opposite, adjacent, and hypotenuse for a given acute angle.
• I can set up the correct trigonometric ratio and solve for an unknown side or angle.
• I can use inverse trig (or inverse reasoning) appropriately to find an angle.
• I can check my answer for reasonableness (angle type and relative side lengths).

Curriculum links

• Trigonometric ratios in right triangles (sine, cosine, tangent) and solving problems
• Applying tools and strategies to solve for unknowns in geometry contexts
• Communicating mathematical thinking using clear steps, diagrams, and reasoning

Lesson structure (30 minutes)

1. 3 min — Hook & quick recall Show one right triangle diagram (no numbers) and ask: “For this acute angle, which side is opposite? Which is adjacent? Which is the hypotenuse?” Students discuss with a partner and share one answer.

2. 6 min — Key concepts mini-teach Introduce:

• Sine: opposite/hypotenuse
• Cosine: adjacent/hypotenuse
• Tangent: opposite/adjacent Emphasize: ratios depend on the chosen acute angle. Model choosing the ratio with a quick worked setup.

3. 8 min — Worked example: missing side Example 1: A right triangle has an acute angle of 30°. The hypotenuse is 10 cm. Find the opposite side.

• Step 1: Identify sides relative to 30° (opposite is the side asked; hypotenuse is given).
• Step 2: Write the ratio: sin(30°) = opposite/10.
• Step 3: Solve: opposite = 10·sin(30°).
• Step 4: Calculate and round appropriately (e.g., to 1 decimal place if needed). Do a second micro-example without computing: “If you know hypotenuse and want adjacent, which ratio uses adjacent and hypotenuse?”

4. 8 min — Guided example: missing angle Example 2: A right triangle has an acute angle that corresponds to an opposite side of 6 m and a hypotenuse of 10 m. Find the angle measure.

• Step 1: Identify the ratio using the given sides: sin(angle) = 6/10.
• Step 2: Use inverse sine: angle = sin⁻¹(6/10).
• Step 3: Interpret the result as an acute angle and round. Remind students: use inverse trig to move from a ratio value back to an angle.

5. 4 min — Student practice (pair or individual) Give 2 short problems on the board or handout (students can choose one side-finding and one angle-finding):

• Problem A (side): Right triangle, cos(θ) = adjacent/hypotenuse, with adjacent = 7.5 cm, hypotenuse = 9.0 cm. Find θ’s tangent? (Teacher notes: this checks understanding of ratios; then students compute an angle if desired.)
• Problem B (angle): Right triangle with opposite = 5, adjacent = 8 for an acute angle. Find the angle θ. Teacher circulates and prompts: “Which angle are you using, and what are opposite/adjacent/hypotenuse for that angle?”

6. 1 min — Exit ticket Each student submits one response: “For a right triangle, if you know opposite and hypotenuse to find an angle, which ratio do you use, and why?” (They answer with a ratio name and a one-sentence reason.)

Resources

• Right-triangle diagram templates (labelled space for hypotenuse, opposite, adjacent)
• Calculator access for inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹)
• Two worked examples printed or displayed
• Practice problem set (2 questions) and a one-question exit ticket
• Student notebooks or worksheets for showing setup, substitution, and final rounding

Assessment

• Formative teacher observation during guided practice: correct side identification and ratio choice
• Short pair/individual check during the two examples: students show ratio setup, not just answers
• Exit ticket: students correctly state the ratio and provide a brief justification related to the chosen acute angle

Differentiation

• Support: Provide a “ratio decision chart” (sin = opp/hyp; cos = adj/hyp; tan = opp/adj) and sentence stems: “For angle θ, the side opposite is…, the hypotenuse is…, so I use…”
• Support: Offer one additional example where only side labels are practised before introducing calculations
• Extension: Include a problem requiring selection between sine, cosine, and tangent when given three sides and an acute angle, followed by a reasonableness check using relative side lengths
• EAL/SEN: Reduce language load by using consistent diagrams, colour-coding opposite/adjacent/hypotenuse, and allowing brief verbal explanations paired with labelled work