Introduction to Trigonometry

Label the triangle, select the ratio, solve the equation. Three steps. Every trigonometry problem in this course follows this pattern.

55–60 min MS-M2 3 MC 3 SA Lesson 4 of 22 Free
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Think First

Two right-angled triangles both have a 35° angle, but one is much larger than the other. Would the ratio of the opposite side to the hypotenuse be the same in both triangles, or different? Why?

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SOHCAHTOA — The Three Ratios

$\sin\theta = \dfrac{O}{H}$
SOH — Opposite over Hypotenuse Rearranged: $O = H\sin\theta$  |  $H = \dfrac{O}{\sin\theta}$
$\cos\theta = \dfrac{A}{H}$
CAH — Adjacent over Hypotenuse Rearranged: $A = H\cos\theta$  |  $H = \dfrac{A}{\cos\theta}$
$\tan\theta = \dfrac{O}{A}$
TOA — Opposite over Adjacent Rearranged: $O = A\tan\theta$  |  $A = \dfrac{O}{\tan\theta}$
Finding an angle: $\theta = \sin^{-1}\!\left(\tfrac{O}{H}\right)$, $\cos^{-1}\!\left(\tfrac{A}{H}\right)$, or $\tan^{-1}\!\left(\tfrac{O}{A}\right)$   |   Check first: sin(90) = 1 confirms degree mode
LABELLING THE TRIANGLE — SOHCAHTOA θ O opp A adj H hyp SOH sinθ = O/H opp ÷ hyp CAH cosθ = A/H adj ÷ hyp TOA tanθ = O/A opp ÷ adj Which ratio to use? O & H → sin  ·  A & H → cos  ·  O & A → tan Identify the two sides involved, then match the ratio

Know

  • The three trig ratios — sine, cosine, tangent — and their abbreviations
  • The SOHCAHTOA memory device
  • How to use $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ on a calculator

Understand

  • Why the ratio of two sides depends only on the angle, not the triangle's size
  • Why selecting the ratio requires identifying the two sides involved
  • Why inverse trig functions find angles rather than sides

Can Do

  • Label opposite, adjacent, and hypotenuse relative to any given angle
  • Select and apply the correct ratio to find an unknown side or angle
  • Use a calculator in degree mode correctly for all trig calculations
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Key Vocabulary

Opposite sideThe side directly across from the reference angle — not touching it
Adjacent sideThe side next to the reference angle that is not the hypotenuse
Sine (sin)The ratio of the opposite side to the hypotenuse for a given angle
Cosine (cos)The ratio of the adjacent side to the hypotenuse for a given angle
Tangent (tan)The ratio of the opposite side to the adjacent side for a given angle
Degree modeCalculator setting required for all HSC trig — verify by checking sin(90) = 1

Misconceptions to Fix

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Wrong: A percentage over 100% is impossible.

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Right: Percentages over 100% are valid and common in contexts like percentage increase, profit margins, and scale factors.

Core Content

01

Why Trigonometry Works

Take any right-angled triangle with a 35° angle. Make it bigger. Make it smaller. As long as that angle stays at 35°, the ratio of any two sides stays exactly the same.

This is the key idea. For any 35° right-angled triangle in the world, opposite ÷ hypotenuse is always the same number. That number is $\sin 35°$.

Trigonometry names and uses these fixed ratios. Once you know an angle, you know all three ratios. Once you know a ratio and a side length, you can find any other side.

Why this is useful: Real-world triangles — ramps, roofs, survey lines, staircases — can be described by an angle and one side. Trig finds the other sides without measuring them physically.
02

Labelling the Triangle — The Most Important Step

Before writing any formula, label the three sides relative to the angle you are working with. This is the step most often skipped — and the one that causes the most errors.

Which side
Opposite the right angle — always the longest side
Directly across from the reference angle (not touching it)
Next to the reference angle, but not the hypotenuse
Does it change?
No — never changes
Yes — changes with the reference angle
Yes — changes with the reference angle

Labelling Process (5 seconds, every time)

  1. Mark the right angle
  2. Circle the reference angle (the one you know or are finding)
  3. Label H opposite the right angle
  4. Label O opposite the circled angle
  5. Label A — the remaining side
Do this on every diagram before writing any formula. It takes 5 seconds and prevents the most common error in the entire trig topic. The ratio selection that follows is then mechanical — just match labels to the table.
LABELLING CHANGES WITH THE REFERENCE ANGLE θ O A H θ at bottom-right θ H O A θ at top-left — O and A swap! H never changes. O and A always do.
03

The Three Ratios — SOHCAHTOA

$$\sin\theta = \frac{O}{H} \qquad \cos\theta = \frac{A}{H} \qquad \tan\theta = \frac{O}{A}$$

SOHCAHTOA is a memory device — read it as three chunks: SOH, CAH, TOA.

Choosing the Right Ratio

Identify which two sides are involved (one known, one unknown), then match to the table:

Ratio to use
Column B

Where the Unknown Sits Determines the Operation

Unknown in numerator?
Yes — $x$ is on top
No — $x$ is in denominator
Solve by
$x = H \times \sin\theta$ (multiply)
$x = \dfrac{O}{\sin\theta}$ (divide)
Algebra beats memory: Never try to remember "multiply or divide" as a rule. Always write the equation and solve algebraically. Two lines of working: rearrange, then evaluate. This earns method marks even if the final calculation goes wrong.
04

Finding an Unknown Angle

When you know two sides and need the angle, use the inverse trig functions.

$$\text{If } \sin\theta = 0.6, \text{ then } \theta = \sin^{-1}(0.6)$$

The inverse function undoes the ratio and gives back the angle. On your calculator: press SHIFT (or 2ND) then the sin/cos/tan key.

Degrees and Minutes

HSC questions sometimes require angles in degrees and minutes instead of decimal degrees.

Convert
$0.7 \times 60 = 42$ min
$0.0724 \times 60 = 4.3 \approx 4$ min
Degrees and minutes
$34°42'$
$28°4'$
Degree mode check — do this first: Type $\sin(90)$. If the answer is 1, you are in degree mode. If not, switch immediately: MODE → DEG. Do this at the start of every trig session and before every exam. Wrong mode = every answer wrong.
05

Common Mistakes

Mistake 1 — Wrong ratio because triangle wasn't labelled first
Guessing the ratio based on the shape of the triangle rather than identifying the sides. Fix: label O, A, H relative to the reference angle on every diagram before writing any formula. 5 seconds. Non-negotiable.
Mistake 2 — Calculator in radian mode
$\sin(32)$ in radian mode gives $-0.5514$ instead of $0.5299$. The answer will be completely wrong. Check: $\sin(90) = 1$ before every trig calculation.
Mistake 3 — Wrong rearrangement when unknown is in the denominator
$\cos 48° = \dfrac{9}{H}$ becomes $H = 9 \times \cos 48°$ (wrong — multiplied instead of divided). Always solve algebraically: multiply both sides by $H$, then divide both sides by $\cos 48°$. Write both lines explicitly.

Worked Examples

06
Worked Example 1
Finding an Unknown Side (Opposite)

Problem

In a right-angled triangle, the reference angle is 32°, the hypotenuse is 15 cm, and $x$ is the opposite side. Find $x$ correct to 2 decimal places.

Step-by-Step Solution

1
Label and identify sides
O = $x$ (unknown), H = 15 cm (known)
→ Sides: O and H → use sin
Label the triangle first: circle 32°, label H opposite the right angle, label O opposite 32°. Two sides involved: opposite and hypotenuse.
07
Worked Example 2
Finding the Hypotenuse

Problem

In a right-angled triangle, the reference angle is 48°, the adjacent side is 9 m, and the hypotenuse is unknown. Find the hypotenuse correct to 2 decimal places.

Step-by-Step Solution

1
Label and identify sides
A = 9 m (known), H = unknown
→ Sides: A and H → use cos
Adjacent and hypotenuse — use cosine.
08
Worked Example 3
Finding an Unknown Angle

Problem

A right-angled triangle has opposite side 7 cm and hypotenuse 11 cm. Find the reference angle $\theta$ correct to the nearest degree.

Step-by-Step Solution

1
Identify sides and ratio
O = 7, H = 11 → use sin
Finding the angle → use $\sin^{-1}$
Opposite and hypotenuse — sine. We know the ratio, need the angle — use the inverse function.
09
Worked Example 4
Angle in Degrees and Minutes

Problem

Find the angle $\theta$ if $\tan\theta = 0.842$. Give your answer in degrees and minutes.

Step-by-Step Solution

1
Apply inverse function
$\theta = \tan^{-1}(0.842) = 40.0876...°$
Given the tan ratio directly — apply $\tan^{-1}$ immediately.
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Trigonometry Ratio Selection and Calculation

For every question: label O/A/H, state the ratio and why, then solve.

Section A — Identify the Correct Ratio (no calculation needed)

1 Reference angle = 40°. Known side = hypotenuse. Unknown = opposite. Which ratio?

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2 Reference angle = 55°. Known side = adjacent. Unknown = hypotenuse. Which ratio?

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3 Reference angle = 28°. Known side = opposite. Unknown = adjacent. Which ratio?

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4 Reference angle = 63°. Known side = hypotenuse. Unknown = adjacent. Which ratio?

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Section B — Find the Unknown Side

5 Reference angle = 35°, hypotenuse = 12 cm. Find the opposite side.

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6 Reference angle = 50°, hypotenuse = 20 m. Find the adjacent side.

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7 Reference angle = 42°, adjacent = 8 cm. Find the hypotenuse.

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8 Reference angle = 67°, opposite = 14 m. Find the hypotenuse.

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9 Reference angle = 38°, adjacent = 10 cm. Find the opposite side.

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10 Reference angle = 22°, opposite = 5 m. Find the adjacent side.

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Section C — Find the Unknown Angle

11 Opposite = 6 cm, hypotenuse = 10 cm. Find $\theta$ to the nearest degree.

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12 Adjacent = 8 m, hypotenuse = 13 m. Find $\theta$ to the nearest degree.

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13 Opposite = 9 cm, adjacent = 4 cm. Find $\theta$ in degrees and minutes.

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Show Answers

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Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

Multiple Choice

1 In a right-angled triangle, the reference angle is $\theta$, the adjacent side is 7 cm, and the hypotenuse is 14 cm. The value of $\theta$ is:

A   30°
B   45°
C   60°
D   90°

? Regarding this topic, 1 In a right-angled triangle, the reference angle is $\theta$, the adjacent side is 7 cm, and the hypotenuse is 14 cm. The value of $\theta$ is:

A     30°
B     45°
C     60°
D     90°
A - Correct!
A — $\cos\theta = A/H = 7/14 = 0.5$. $\theta = \cos^{-1}(0.5) = 30°$.

2 A right-angled triangle has a reference angle of 55° and adjacent side of 10 m. The opposite side is closest to:

A   5.74 m
B   8.19 m
C   12.21 m
D   14.28 m

? Regarding this topic, 2 A right-angled triangle has a reference angle of 55° and adjacent side of 10 m. The opposite side is closest to:

A     5.74 m
B     8.19 m
C     12.21 m
D     14.28 m
D - Correct!
D — $\tan 55° = O/10$. $O = 10 \times \tan 55° = 10 \times 1.4281 = 14.28$ m.

3 Which expression correctly gives the hypotenuse $H$ in terms of $\theta$ and the opposite side $O$?

A   $H = O \times \sin\theta$
B   $H = \sin\theta / O$
C   $H = O / \sin\theta$
D   $H = O \times \cos\theta$

? Regarding this topic, 3 Which expression correctly gives the hypotenuse $H$ in terms of $\theta$ and the opposite side $O$?

A     $H = O \times \sin\theta$
B     $H = \sin\theta / O$
C     $H = O / \sin\theta$
D     $H = O \times \cos\theta$
C - Correct!
C — $\sin\theta = O/H$ → $H = O/\sin\theta$. Multiply both sides by $H$, then divide by $\sin\theta$.

Short Answer

10

SA 4 2 marks Find the length of side $x$ in a right-angled triangle where the reference angle is 41°, $x$ is the adjacent side, and the hypotenuse is 18 cm. Give your answer correct to 2 decimal places.

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11

SA 5 3 marks A right-angled triangle has opposite side 8 m and hypotenuse 17 m.

(a) Write down the trigonometric ratio that connects these two sides.  (1 mark)

(b) Find the reference angle $\theta$ correct to the nearest minute.  (2 marks)

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12

SA 6 4 marks A ramp rises from ground level to a platform. The ramp makes an angle of 15° with the horizontal. The horizontal distance is 6 m.

(a) Draw a labelled diagram showing this situation.  (1 mark)

(b) Find the length of the ramp (hypotenuse) correct to 2 decimal places.  (1 mark)

(c) Find the height of the platform correct to 2 decimal places.  (2 marks)

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Introduction to Trigonometry

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