Mathematics Standard • Year 11 • Module 2 • Lesson 4

Introduction to Trigonometry

Build fluency in SOHCAHTOA: label the triangle, choose the correct ratio, solve for the unknown side or angle.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete each SOHCAHTOA ratio:

sin θ = _______ / _______ cos θ = _______ / _______ tan θ = _______ / _______

Q1.2 Match the side type to its definition (write A, B or C beside each):

______ Opposite (O) ______ Adjacent (A) ______ Hypotenuse (H)

A. Opposite the right angle (longest side) B. Directly across from the reference angle C. Next to the reference angle, not the hypotenuse

Q1.3 Calculator check: type sin(90). If the answer is ______ you are in degree mode. (One-word answer.)

Stuck? Revisit lesson § SOHCAHTOA — The Three Ratios and § Calculator Degree Mode Check.

2. Worked example — find an unknown side

Follow each line — every step earns a method mark.

Problem. A right-angled triangle has reference angle 32°, hypotenuse 15 cm. Find the opposite side, to 2 d.p.

Step 1 — Label and identify the sides involved.

O = x (unknown), H = 15 cm (known). Sides: O and H → use SIN.

Reason: SOH — sin uses opposite and hypotenuse.

Step 2 — Write the equation.

sin 32° = x / 15

Reason: substitute θ = 32°, O = x, H = 15.

Step 3 — Rearrange (unknown in numerator → multiply).

x = 15 × sin 32°

Reason: multiply both sides by 15 to isolate x.

Step 4 — Evaluate and state with units.

x = 15 × 0.52992... = 7.9488...

x = 7.95 cm (to 2 d.p.)

3. Faded example — find an unknown angle

A right-angled triangle has opposite side 7 cm and hypotenuse 11 cm. Find the reference angle θ to the nearest degree. Fill in every blank. 3 marks

Step 1 — Sides involved: O and ______ → use the ____________ ratio.

Step 2 — Equation: sin θ = ______ / ______

Step 3 — Apply inverse function: θ = sin⁻¹( ______ / ______ )

Step 4 — Evaluate: θ = ______ ° (to nearest degree)

Sense check: 7/11 ≈ 0.64 — an angle whose sine is 0.64 must be less than 90°.

Stuck? Revisit lesson § Worked Example 3 — Finding an Unknown Angle. Calculator: SHIFT then sin key for sin⁻¹.

4. Graduated practice — choose the ratio, then solve

For every question: identify the two sides involved (and so the ratio), write the equation, then solve.

Foundation — choose the correct ratio only (4 questions)

Q	Sides involved (one known, one unknown)	Ratio (sin / cos / tan)
4.1 1	Hypotenuse known, opposite unknown.
4.2 1	Adjacent known, hypotenuse unknown.
4.3 1	Opposite known, adjacent unknown.
4.4 1	Hypotenuse known, adjacent unknown.

Standard — find the unknown side (6 questions, 2 d.p.)

4.5 θ = 35°, hypotenuse = 12 cm. Find the opposite side. 2 marks

4.6 θ = 50°, hypotenuse = 20 m. Find the adjacent side. 2 marks

4.7 θ = 42°, adjacent = 8 cm. Find the hypotenuse. 2 marks

4.8 θ = 67°, opposite = 14 m. Find the hypotenuse. 2 marks

4.9 θ = 38°, adjacent = 10 cm. Find the opposite side. 2 marks

4.10 θ = 22°, opposite = 5 m. Find the adjacent side. 2 marks

Extension — find the unknown angle (2 questions)

4.11 Opposite = 6 cm, hypotenuse = 10 cm. Find θ to the nearest degree. 2 marks

4.12 Opposite = 9 cm, adjacent = 4 cm. Find θ in degrees and minutes. 3 marks

Stuck on 4.12? Find the decimal degree first, then multiply the decimal part by 60 to get minutes.

5. Self-check the easy 3

Tick the first three once you've checked your method.

I typed sin(90) before starting and got 1 — calculator confirmed in degree mode.

For 4.7 (unknown is the hypotenuse, in the denominator) I DIVIDED by cos 42°, not multiplied.

For 4.11 my angle answer is between 0° and 90° — sense check on inverse trig.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — SOHCAHTOA

sin θ = O / H. cos θ = A / H. tan θ = O / A.

Q1.2 — Side definitions

Opposite (O) → B. Adjacent (A) → C. Hypotenuse (H) → A.

Q1.3 — Degree mode check

If sin(90) = 1, you are in degree mode. (If in radian mode, sin(90) ≈ 0.894.)

Q3 — Faded angle example

Step 1: O and H → use sin. Step 2: sin θ = 7/11. Step 3: θ = sin⁻¹(7/11). Step 4: θ = 39.52...° ≈ 40°.

Q4.1 — H known, O unknown

O and H → sin.

Q4.2 — A known, H unknown

A and H → cos.

Q4.3 — O known, A unknown

O and A → tan.

Q4.4 — H known, A unknown

A and H → cos.

Q4.5 — θ = 35°, H = 12, find O

sin 35° = O / 12 → O = 12 × sin 35° = 6.88 cm.

Q4.6 — θ = 50°, H = 20, find A

cos 50° = A / 20 → A = 20 × cos 50° = 12.86 m.

Q4.7 — θ = 42°, A = 8, find H

cos 42° = 8 / H → H = 8 / cos 42° = 10.76 cm. Sense check: H > 8 ✓ (hypotenuse longest).

Q4.8 — θ = 67°, O = 14, find H

sin 67° = 14 / H → H = 14 / sin 67° = 15.21 m.

Q4.9 — θ = 38°, A = 10, find O

tan 38° = O / 10 → O = 10 × tan 38° = 7.81 cm.

Q4.10 — θ = 22°, O = 5, find A

tan 22° = 5 / A → A = 5 / tan 22° = 12.37 m.

Q4.11 — O = 6, H = 10, find θ

sin θ = 6/10 = 0.6. θ = sin⁻¹(0.6) = 36.87...° ≈ 37°.

Q4.12 — O = 9, A = 4, find θ in degrees and minutes

tan θ = 9/4 = 2.25. θ = tan⁻¹(2.25) = 66.037...°. Decimal part: 0.037 × 60 = 2.22 ≈ 2 min. θ = 66°2'.