Mathematics • Year 10 • Unit 3 • Lesson 1
Introduction to Trigonometric Ratios — Skill Drill
Build fluency with SOH CAH TOA from Lesson 1: label opposite (O), adjacent (A), hypotenuse (H) relative to a marked angle θ, write the three trig ratios as fractions, and recall the exact values for 30°, 45° and 60° without a calculator.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. A right-angled triangle has the right angle at the top-left corner. The angle θ sits at the bottom-left. The hypotenuse is 13 cm, the side opposite θ is 5 cm, and the side adjacent to θ is 12 cm. Write the three trig ratios for θ as fractions in simplest form.
Step 1 — Identify the three sides.
H = 13 cm, O = 5 cm, A = 12 cm
Reason: H is opposite the right angle (always the longest). O is across from θ. A is next to θ but not the hypotenuse.
Step 2 — Apply SOH for sine.
sin θ = O / H = 5 / 13
Reason: SOH says Sine = Opposite ÷ Hypotenuse. The fraction 5/13 is already in simplest form.
Step 3 — Apply CAH for cosine.
cos θ = A / H = 12 / 13
Reason: CAH says Cosine = Adjacent ÷ Hypotenuse.
Step 4 — Apply TOA for tangent.
tan θ = O / A = 5 / 12
Reason: TOA says Tangent = Opposite ÷ Adjacent. Tangent never involves the hypotenuse.
Answer: sin θ = 5/13, cos θ = 12/13, tan θ = 5/12.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. A right-angled triangle has hypotenuse 25, the side opposite θ is 7, and the side adjacent to θ is 24. Write sin θ, cos θ and tan θ as fractions in simplest form.
Step 1 — Label the sides:
H = ______ , O = ______ , A = ______
Step 2 — Use SOH for sine:
sin θ = ______ / ______ = ______
Step 3 — Use CAH for cosine:
cos θ = ______ / ______ = ______
Step 4 — Use TOA for tangent:
tan θ = ______ / ______ = ______
Step 5 — Sanity check. The hypotenuse must be the longest side. Is 25 ≥ both 7 and 24? ____ ✓
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — labelling and ratio writing
3.1 A right triangle has H = 17, O = 8, A = 15 (relative to angle θ). Write sin θ as a fraction in simplest form. 1 mark
3.2 For the same triangle (H = 17, O = 8, A = 15), write cos θ as a fraction. 1 mark
3.3 For the same triangle (H = 17, O = 8, A = 15), write tan θ as a fraction. 1 mark
3.4 Recall the exact value of sin 30°, as a simple fraction. 1 mark
Standard — combine the exact-value table
3.5 Without a calculator, evaluate: cos 60° + sin 30°. Give your answer as a single fraction. 2 marks
3.6 A right-angled triangle has sides 9, 12, 15 (with 15 as the hypotenuse). The angle θ is opposite the side of length 9. Write all three trig ratios for θ in simplest form. 2 marks
Extension — push your thinking
3.7 In a right-angled triangle, sin θ = 3/5 and the hypotenuse is 20 cm. Find the length of the side opposite θ, and the length of the side adjacent to θ. (Hint: use the 3-4-5 triangle.) 3 marks
3.8 Without a calculator, evaluate: (sin 45°) × (cos 45°). Give your answer as a simple fraction. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (7-24-25 triangle)
Step 1: H = 25, O = 7, A = 24.
Step 2: sin θ = 7 / 25 = 7/25.
Step 3: cos θ = 24 / 25 = 24/25.
Step 4: tan θ = 7 / 24 = 7/24.
Step 5: 25 is the longest side. ✓ (This is a Pythagorean triple: 7² + 24² = 49 + 576 = 625 = 25².)
3.1 — sin θ for the 8-15-17 triangle
sin θ = O / H = 8/17. Already in simplest form.
3.2 — cos θ for the 8-15-17 triangle
cos θ = A / H = 15/17.
3.3 — tan θ for the 8-15-17 triangle
tan θ = O / A = 8/15.
3.4 — sin 30°
sin 30° = 1/2. (From the 30-60-90 exact-value table.)
3.5 — cos 60° + sin 30°
cos 60° = 1/2 and sin 30° = 1/2. Sum = 1/2 + 1/2 = 1.
This is one of the patterns in the exact-value table: sin 30° = cos 60° because 30° and 60° are complementary angles.
3.6 — 9-12-15 triangle (θ opposite 9)
H = 15, O = 9, A = 12.
sin θ = 9/15 = 3/5.
cos θ = 12/15 = 4/5.
tan θ = 9/12 = 3/4.
This is a 3-4-5 triangle scaled up by 3. Always simplify the fractions.
3.7 — sin θ = 3/5, H = 20 cm
sin θ = O / H, so O = H × sin θ = 20 × (3/5) = 12 cm.
Using the 3-4-5 ratio, A = 20 × (4/5) = 16 cm.
Check with Pythagoras: 12² + 16² = 144 + 256 = 400 = 20². ✓
3.8 — (sin 45°) × (cos 45°)
sin 45° × cos 45° = (1/√2) × (1/√2) = 1/(√2 × √2) = 1/2.
The exact-value table makes the calculator unnecessary — and this expression appears constantly in later trig identities.