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.