Label the triangle, select the ratio, solve the equation. Three steps. Every trigonometry problem in this course follows this pattern.
55–60 minMS-M23 MC3 SALesson 4 of 22Free
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Choose how you work: type answers on screen, or work in your book.
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?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
TOA — Opposite over AdjacentRearranged: $O = A\tan\theta$ | $A = \dfrac{O}{\tan\theta}$
Finding an angle: $\theta = \sin^{-1}\!\left(\tfrac{O}{H}\right)$, $\cos^{-1}\!\left(\tfrac{A}{H}\right)$, or $\tan^{-1}\!\left(\tfrac{O}{A}\right)$
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Check first: sin(90) = 1 confirms degree mode
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: You can find an angle in a right-angled triangle if you only know the other two angles
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Right: Trigonometric ratios require at least one side length and one angle, or two side lengths. You cannot determine unknown sides from angles alone. Also: SOH means sin = opposite/hypotenuse, where "opposite" and "hypotenuse" are labelled relative to the angle in question.
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.
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Trigonometry Ratio Selection and Calculation
For every question: label O/A/H, state the ratio and why, then solve.
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)
Mark the right angle
Circle the reference angle (the one you know or are finding)
Label H opposite the right angle
Label O opposite the circled angle
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.
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.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
04
Finding an Unknown Angle
When you know two sides and need the angle, use the inverse trig functions.
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.
Check Your Understanding
Try a quick calculation: write down one example from this section and work through it.
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.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
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.
$x$ is in the numerator of the right side, so multiply both sides by 15.
4
Evaluate and state answer $x = 15 \times 0.52992... = 7.9488...$ $x = 7.95\text{ cm}$
Evaluate $\sin 32°$ using the calculator (degree mode). Round to 2 d.p. at the final step only.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
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.
2
Write the equation $\cos 48° = \dfrac{9}{H}$
$\cos\theta = A/H$. The unknown $H$ is in the denominator.
Multiply both sides by $H$. Then divide both sides by $\cos 48°$. Write both algebra lines — they earn method marks.
4
Evaluate and state answer $H = \dfrac{9}{0.66913...} = 13.4506...$ $H = 13.45\text{ m}$
Common error: $H = 9 \times \cos 48° = 6.02$ m — that is the adjacent multiplied back, which gives a value smaller than the adjacent. Impossible since H is the hypotenuse (longest side). Check: answer must be larger than 9 m ✓
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
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.
2
Set up $\sin\theta = \dfrac{7}{11}$ $\theta = \sin^{-1}\!\left(\dfrac{7}{11}\right)$
Leave $\frac{7}{11}$ as a fraction inside the inverse function — better precision than converting to 0.6364 first.
3
Evaluate and round $\theta = 39.52...° \approx 40°$
Round to the nearest degree as instructed. 39.52° rounds up to 40°.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
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.
2
Extract whole degrees $40°$ and decimal part $0.0876°$
The whole number part gives the degrees. Keep the decimal separately for the minutes conversion.
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 — $\tan 55° = O/10$. $O = 10 \times \tan 55° = 10 \times 1.4281 = 14.28$ 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 $H = O \times \cos\theta$
C — $\sin\theta = O/H$ → $H = O/\sin\theta$. Multiply both sides by $H$, then divide by $\sin\theta$.
Short Answer
10
SA 42 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.
Working space in book
Saved
11
SA 53 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)
Working space in book
Saved
12
SA 64 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)