Use Pythagoras' theorem and SOHCAHTOA to find unknown sides and angles in right-angled triangles. Apply these to real-world problems involving distance, height, and angles of elevation/depression.
What You'll Learn
Apply Pythagoras' theorem to find missing sides in right-angled triangles
Label triangles with H, O, A relative to a given angle
Choose the correct trig ratio (sin, cos, tan) for any problem
Calculate unknown sides using SOHCAHTOA
Calculate unknown angles using inverse trig functions
Solve real-world problems: angles of elevation/depression, bearings
Calculate surface area and volume of composite 3D shapes
Key Vocabulary
Term
Definition
Hypotenuse (H)
The longest side; always opposite the right angle
Opposite (O)
The side across from the reference angle
Adjacent (A)
The side next to the reference angle (not the hypotenuse)
Angle of elevation
The angle measured upward from the horizontal
Angle of depression
The angle measured downward from the horizontal
Composite shape
A shape made by combining two or more basic shapes
Cross-section
The 2D shape you see when you slice through a 3D shape
Mathematics · Topic 1.3
Pythagoras' Theorem
Pythagoras' theorem relates the three sides of a right-angled triangle. It works ONLY in right-angled triangles.
Pythagoras' Theorem
a² + b² = c² where c is the hypotenuse
Finding the Hypotenuse
When you know the two shorter sides and need the longest side:
Example: A right-angled triangle has sides 5 cm and 12 cm. Find the hypotenuse.
Common Mistake: Don't assume the hypotenuse is always labelled c! The hypotenuse is always the longest side and always opposite the right angle. Read the diagram carefully.
Mathematics · Topic 1.3
SOHCAHTOA — The Trig Ratios
SOHCAHTOA is a mnemonic for the three trigonometric ratios. They connect angles to side lengths in right-angled triangles.
Label H, O, A relative to angle θ before choosing a trig ratio
Ratio
Full Name
Formula
Mnemonic
sin θ
Sine
Opposite / Hypotenuse
Sin = Opp / Hyp
cos θ
Cosine
Adjacent / Hypotenuse
Cos = Adj / Hyp
tan θ
Tangent
Opposite / Adjacent
Tan = Opp / Adj
Step-by-Step Process
Label the triangle: identify the angle θ, then mark H (opposite the right angle), O (opposite θ), and A (next to θ, not H).
Identify what you know and what you need. Write down the two relevant sides.
Choose the correct ratio: if you have O and H, use sin. If A and H, use cos. If O and A, use tan.
Set up the equation and solve for the unknown.
Check your answer makes sense (the hypotenuse must be the longest side).
Choosing the Right Ratio
You Know
You Need
Use
Angle + Hypotenuse
Opposite
sin θ = O/H → O = H × sin θ
Angle + Hypotenuse
Adjacent
cos θ = A/H → A = H × cos θ
Angle + Adjacent
Opposite
tan θ = O/A → O = A × tan θ
Angle + Opposite
Hypotenuse
sin θ = O/H → H = O / sin θ
Angle + Adjacent
Hypotenuse
cos θ = A/H → H = A / cos θ
Angle + Opposite
Adjacent
tan θ = O/A → A = O / tan θ
Key Angle Values
θ
sin θ
cos θ
tan θ
30°
0.5
0.866
0.577
45°
0.707
0.707
1
60°
0.866
0.5
1.732
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Critical Rule: ALWAYS label the triangle with H, O, and A relative to the given angle BEFORE choosing a ratio. If you change which angle you're working with, O and A swap positions. The most common trigonometry error is using the wrong ratio.
Mathematics · Topic 1.3
Finding Unknown Sides
When you know an angle and one side, you can find any other side using SOHCAHTOA.
Example 1: θ = 35°, H = 12 cm. Find O.
Worked Example — Finding a Side
Have H, need O → use sin: sin θ = O/Hchoose ratio
sin 35° = O/12 → O = 12 × sin 35°rearrange
O = 12 × 0.5736evaluate
O = 6.88 cm
Example 2: θ = 50°, A = 8 cm. Find H.
Worked Example — Finding the Hypotenuse
Have A, need H → use cos: cos θ = A/Hchoose ratio
cos 50° = 8/H → H = 8cos 50° = 80.6428rearrange & substitute
H = 12.44 cm
Example 3: θ = 62°, A = 5 cm. Find O.
Worked Example — Finding a Side
Have A, need O → use tan: tan θ = O/Achoose ratio
tan 62° = O/5 → O = 5 × tan 62° = 5 × 1.8807rearrange & evaluate
O = 9.40 cm
Example 4: θ = 28°, O = 7 cm. Find A.
Worked Example — Finding a Side
Have O, need A → use tan: tan θ = O/Achoose ratio
tan 28° = 7/A → A = 7tan 28° = 70.5317rearrange & substitute
A = 13.17 cm
⚠
Check: After solving, verify that H is the longest side. If your answer for H is shorter than another side, you've made an error. Also ensure your calculator is in degree mode, not radians.
Mathematics · Topic 1.3
Finding Unknown Angles
When you know two sides, use the inverse trig functions (sin¹, cos¹, tan¹) to find the angle.
Angle of elevation Look UP from horizontal to an object above you. The angle is between the horizontal and your line of sight.
Angle of depression Look DOWN from horizontal to an object below you. The angle is between the horizontal and your line of sight.
Example: A person stands 30 m from a building. The angle of elevation to the top is 55°. Find the height of the building.
Worked Example — Angle of Elevation
Height is O, ground distance is A = 30 m → use tanlabel triangle
tan 55° = O/30 → O = 30 × tan 55°rearrange
O = 30 × 1.4281evaluate
O = 42.8 m
⚠
Remember: Angle of elevation from point A to B = angle of depression from B to A (they are alternate angles between parallel horizontal lines).
Mathematics · Topic 1.3
Worked Examples
Multi-step problems combining Pythagoras and trigonometry.
EXAMPLE 1A ladder 5 m long leans against a wall. The foot of the ladder is 2 m from the base of the wall. Find: (a) how high up the wall it reaches, (b) the angle it makes with the ground.
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Full Solution
(a) Using Pythagoras:
H = 5 m (ladder = hypotenuse), A = 2 m (ground distance)
5² = 2² + h² → 25 = 4 + h² → h² = 21 → h = √21 ≈ 4.58 m
(b) Using trigonometry:
cos θ = A/H = 2/5 = 0.4
θ = cos−1(0.4) = 66.4°
EXAMPLE 2From the top of a 40 m cliff, the angle of depression to a boat is 32°. How far is the boat from the base of the cliff?
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Full Solution
The angle of depression from the cliff top = the angle of elevation from the boat = 32° (alternate angles).
O = 40 m (cliff height), A = distance to boat.
tan 32° = O/A = 40/A
A = 40 / tan 32° = 40 / 0.6249 = 64.0 m
EXAMPLE 3An isosceles triangle has two equal sides of 10 cm and a base of 12 cm. Find the height and area.
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Full Solution
The height of an isosceles triangle bisects the base, creating two right-angled triangles.
Each right triangle: H = 10 cm, base = 12/2 = 6 cm.
Height: h² + 6² = 10² → h² = 100 − 36 = 64 → h = 8 cm
Area: A = ½ × base × height = ½ × 12 × 8 = 48 cm²
EXAMPLE 4A 3D box has dimensions 6 cm × 8 cm × 10 cm. Find the angle that the space diagonal makes with the base.
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Full Solution
Step 1: Find the base diagonal: dbase² = 6² + 8² = 36 + 64 = 100 → dbase = 10 cm
Step 2: The space diagonal, base diagonal, and height form a right triangle. Space diagonal is the hypotenuse, height = 10, base diagonal = 10.
dspace² = 10² + 10² = 200 → dspace = √200 = 10√2 ≈ 14.14 cm
Step 3: Angle with base: tan θ = height / base diagonal = 10/10 = 1
θ = tan−1(1) = 45°
EXAMPLE 5A cone has a slant height of 13 cm and a radius of 5 cm. Find the perpendicular height and the volume.
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Full Solution
Height: The radius, height, and slant height form a right triangle.
13² = 5² + h² → 169 = 25 + h² → h² = 144 → h = 12 cm
JUSTIFYA student says: "sin 30° = cos 60°." Is this correct? Justify your answer.
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Model Answer
Yes, this is correct. In a 30-60-90 triangle, the side opposite 30° is the same as the side adjacent to 60°. So sin 30° = O/H = the same fraction as cos 60° = A/H (where A for 60° is the same side as O for 30°).
Both equal 1/2 = 0.5. In general, sin θ = cos(90° − θ).
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer.
State Pythagoras' theorem.
a² + b² = c² where c is the hypotenuse (longest side, opposite the right angle).
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What does SOH stand for?
Sin = Opposite / Hypotenuse
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What does CAH stand for?
Cos = Adjacent / Hypotenuse
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What does TOA stand for?
Tan = Opposite / Adjacent
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How do you find an angle from two sides?
Use inverse trig: θ = sin−1(O/H), cos−1(A/H), or tan−1(O/A)
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What is the 3-4-5 triple?
3² + 4² = 9 + 16 = 25 = 5². Multiples also work: 6-8-10, 9-12-15, etc.
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sin 30° = ?
0.5
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cos 60° = ?
0.5 (same as sin 30°, because sin θ = cos(90° − θ))
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tan 45° = ?
1 (because in a 45-45-90 triangle, opposite = adjacent)
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What is the first step before using SOHCAHTOA?
Label the triangle: identify H, O, and A relative to the given angle.
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What is the angle of elevation?
The angle measured upward from the horizontal to an object above you.
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Volume of a cone?
V = (1/3)πr²h
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Surface area of a cylinder?
SA = 2πr² + 2πrh (two circles + curved surface)
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Space diagonal of a box (l, w, h)?
d = √(l² + w² + h²)
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Which is the hypotenuse?
The longest side of a right-angled triangle, always opposite the 90° angle.