Pythagoras' Theorem & Trigonometry
Trig Ratio Definitions (SOH CAH TOA)
Draw a line from each trigonometric ratio to its definition relative to angle θ in a right-angled triangle.
Identify the Sides
For a right-angled triangle with the right angle at C and given angle θ at A, circle the correct label for each side.
The side opposite angle A is:
The side adjacent to angle A (not the hypotenuse) is:
The longest side, opposite the right angle, is:
If you change the reference angle to B, the side BC becomes:
Which Ratio to Use?
For each pair of known/wanted sides, circle the correct trig ratio to use.
You know the opposite and hypotenuse. Use:
You know the adjacent and hypotenuse. Use:
You know the opposite and adjacent. Use:
You want to find the hypotenuse and know the opposite side and the angle. Use:
Special Angles — Exact Values
Draw a line from each trigonometric expression to its exact value.
Pythagoras' Theorem Formula
Circle the correct statement of Pythagoras' theorem and related facts.
Pythagoras' theorem states:
Pythagoras' theorem applies to:
In c² = a² + b², the letter c always represents:
Find the Hypotenuse — Simple Cases
Circle the correct length of the hypotenuse for each right-angled triangle.
Legs: 3 and 4. Hypotenuse = ?
Legs: 5 and 12. Hypotenuse = ?
Legs: 8 and 15. Hypotenuse = ?
Legs: 6 and 8. Hypotenuse = ?
Find a Shorter Side — Simple Cases
Circle the correct length of the missing shorter side.
Hypotenuse = 13, one leg = 5. Other leg = ?
Hypotenuse = 10, one leg = 6. Other leg = ?
Hypotenuse = 25, one leg = 7. Other leg = ?
Hypotenuse = 15, one leg = 9. Other leg = ?
Pythagorean Triples vs Non-Triples
Sort each set of three numbers: is it a Pythagorean triple (a² + b² = c²) or not?
Trig Equation Setup — Height Problems
Circle the correct trig equation to find the unknown height h.
Find the height of a tree if you stand 20 m away and the angle of elevation is 35°
A kite string is 80 m long at an angle of 55° to the ground. The kite's height is:
From the top of a 30 m cliff, the angle of depression to a boat is 40°. The horizontal distance to the boat is:
Trig Equation Setup — Ladder Problems
A ladder of known length leans against a wall. Circle the correct equation.
A 5 m ladder makes a 70° angle with the ground. How high up the wall does it reach?
A 6 m ladder reaches 5.5 m up a wall. The angle with the ground is:
A ladder makes a 65° angle with the ground and its base is 1.2 m from the wall. The ladder length is:
Find the Hypotenuse Using Trig
Circle the correct calculation for the hypotenuse.
Opposite = 7, angle = 30°. Hypotenuse = ?
Adjacent = 10, angle = 45°. Hypotenuse = ?
Opposite = 12, angle = 60°. Hypotenuse = ?
Find the Opposite Side Using Trig
Circle the correct calculation for the opposite side.
Hypotenuse = 20, angle = 35°. Opposite = ?
Adjacent = 15, angle = 50°. Opposite = ?
Hypotenuse = 9, angle = 60°. Opposite = ?
Find the Adjacent Side Using Trig
Circle the correct calculation for the adjacent side.
Hypotenuse = 18, angle = 40°. Adjacent = ?
Opposite = 11, angle = 55°. Adjacent = ?
Hypotenuse = 25, angle = 30°. Adjacent = ?
Find an Angle Using Inverse Trig
Circle the correct angle (to the nearest degree) for each triangle.
Opposite = 5, hypotenuse = 10. Angle = ?
Adjacent = 4, hypotenuse = 8. Angle = ?
Opposite = 7, adjacent = 7. Angle = ?
Opposite = 6, adjacent = 10. Angle = ?
Multi-Step Trig — River Crossing
Put the steps in order to find the distance across a river using trigonometry.
Pythagoras in 3D — Diagonal of a Box
Put the steps in order to find the space diagonal of a box with length 3 m, width 4 m, and height 12 m.
Scenario to Trig Ratio
Draw a line from each scenario to the trig ratio you would use to solve it.
Angle of Elevation vs Depression
Circle the correct statement about angles of elevation and depression.
The angle of elevation is measured:
The angle of depression is measured:
If the angle of elevation from A to B is 35°, the angle of depression from B to A is:
From the top of a 50 m tower, the angle of depression to a car is 25°. The horizontal distance to the car is:
Flagpole Shadow Problem
Solve this practical problem. Draw a diagram and show all working.
A 12 m flagpole casts a shadow 7 m long. (a) Calculate the angle of elevation of the sun. (b) Later in the day, the angle of elevation drops to 30°. Calculate the new shadow length. (c) At what angle of elevation would the shadow be exactly the same length as the flagpole?
Ship Bearing and Distance
Solve this navigation problem. Draw a diagram and show all working.
A ship sails 15 km due east, then 20 km due north. (a) Calculate the direct distance from the starting point to the ship's final position. (b) Calculate the bearing of the ship from its starting point. (c) If the ship then sailed directly back to the start, on what bearing would it travel?
Building Height from Two Positions
Solve this surveying problem. Draw a diagram and show all working.
From point A, the angle of elevation to the top of a building is 40°. You walk 30 m closer to the building to point B, where the angle of elevation is now 55°. Let h be the height of the building and d be the horizontal distance from B to the building. Set up two equations: h = (d + 30) × tan 40° and h = d × tan 55°. Solve for d and hence find h.
Area of a Triangle Using Trigonometry
Use the formula Area = ½ab sin C to solve these problems. Show all working.
A triangle has two sides of length 8 cm and 11 cm with an included angle of 50°. Calculate the area of the triangle using the formula Area = ½ × a × b × sin C.
A triangular garden has sides of 6 m and 9 m with an included angle of 120°. Calculate the area. (Hint: sin 120° = sin 60° = √3 ÷ 2 ≈ 0.866.)
3D Pythagoras — Space Diagonal
Apply Pythagoras' theorem in three dimensions. Show all working.
A room is 5 m long, 4 m wide, and 3 m high. (a) Calculate the length of the diagonal across the floor. (b) Calculate the space diagonal (from one bottom corner to the opposite top corner). (c) A spider at one bottom corner wants to walk to the opposite top corner along the walls. If it goes across the floor and then up the wall, what is the shortest path along the surfaces? (Hint: 'unfold' two walls.)
Navigation and Bearing Problem
Solve this navigation problem using trigonometry. Show all working.
A bushwalker leaves camp and walks 4 km on a bearing of 060°, then 3 km on a bearing of 150°. (a) How far east and north is the walker from camp after the first leg? (Use: east = 4 sin 60° ≈ 3.46 km, north = 4 cos 60° = 2 km.) (b) How far east and south does the second leg take the walker? (Use: east = 3 sin 30° = 1.5 km, south = 3 cos 30° ≈ 2.60 km.) (c) Calculate the walker's final position relative to camp and the direct distance back.
True or False — Pythagoras & Trig Properties
Circle TRUE or FALSE for each statement.
In a right-angled triangle, the hypotenuse is always the longest side
sin θ can be greater than 1 for angles in a right triangle
tan 45° = 1 because the opposite and adjacent sides are equal
Pythagoras' theorem works for all triangles, not just right-angled ones
As an angle increases from 0° to 90°, sin θ increases from 0 to 1
The angle of elevation from A to B equals the angle of depression from B to A
Surveying Problem — Multiple Triangles
Solve this multi-step surveying problem. Draw a diagram and show all working.
A surveyor needs to find the distance between two points, P and Q, on opposite sides of a lake. She stands at point R, which is 120 m from P and 85 m from Q. She measures angle PRQ = 72°. Using the cosine rule (c² = a² + b² − 2ab cos C), calculate the distance PQ. Then calculate the area of triangle PRQ using the formula Area = ½ × a × b × sin C.
Measure Heights Using Trigonometry
Use trigonometry to measure the height of tall objects around your home or neighbourhood.
- 1Use a protractor app (clinometer) on a phone to measure the angle of elevation to the top of a tall tree, building, or flagpole. Measure your distance from the base, then use h = d × tan θ to calculate the height. Remember to add your eye height!
- 2On a sunny day, measure your own height and shadow length to find the sun's angle of elevation (tan θ = height ÷ shadow). Then measure the shadow of a tall object and use the same angle to calculate its height.
- 3Stand at two different distances from a tall object and measure the angle of elevation each time. Use the two-position method from the worksheet to calculate the height without knowing the base distance.
Verify Pythagoras with Household Objects
Test Pythagoras' theorem using real measurements at home.
- 1Measure the length, width, and diagonal of a rectangular table, door, or TV screen. Check whether diagonal² ≈ length² + width². How close is your result?
- 2Cut a piece of string to exactly 12 units (e.g. 60 cm marked at every 5 cm). Form a 3-4-5 triangle with the string. Verify that the angle opposite the longest side is 90° using a set square or the corner of a book.
- 3Measure the three dimensions of a shoebox (length, width, height). Calculate the space diagonal using D = √(l² + w² + h²). Then measure the actual diagonal with a tape measure and compare.
Trigonometric Ratios — Define and Apply
Define and apply sin, cos, and tan to right triangles.
In a right triangle with hypotenuse 13 cm, one leg 5 cm, and other leg 12 cm, find: (a) sin, cos, tan of the smallest acute angle (b) sin, cos, tan of the largest acute angle (c) The angle values in degrees using inverse trig
In a right triangle with one angle of 35° and hypotenuse 20 cm, find the lengths of both legs correct to 2 decimal places. Show which ratio you used and why.
Trigonometric Ratio Definitions
Draw a line from each trig ratio to its definition (relative to angle θ in a right triangle).
Find the Missing Side — Correct Setup
Circle the correct trigonometric equation to find the missing side x.
Angle = 40°, hypotenuse = 15 cm, find opposite side x:
Angle = 55°, adjacent = 8 cm, find opposite side x:
Angle = 28°, adjacent = 12 cm, find hypotenuse x:
Angle = 62°, hypotenuse = 25 cm, find adjacent side x:
Angles of Elevation and Depression
Apply trigonometry to angles of elevation and depression.
A person stands 40 m from the base of a building. They look up at the top at an angle of elevation of 52°. How tall is the building? Draw a diagram before calculating.
A lighthouse is 60 m tall. The keeper sees a boat at an angle of depression of 18°. How far is the boat from the base of the lighthouse? Explain why the angle of depression equals the alternate angle of elevation.
An aeroplane at altitude 3000 m starts descending at an angle of depression of 6°. Over what horizontal distance does it descend before reaching the runway? Give the answer in km.
Classify: Pythagoras or Trigonometry?
Sort each problem: solved using Pythagoras' theorem or trigonometry (or both).
Special Triangles — Exact Values
Circle the exact trigonometric value for each ratio.
sin 30°
cos 60°
tan 45°
sin 60°
cos 30°
tan 60°
Three-Dimensional Trigonometry
Apply Pythagoras and trigonometry to 3D problems.
A room is 8 m long, 5 m wide, and 3 m high. Calculate: (a) The diagonal of the floor. (b) The space diagonal from one corner of the floor to the opposite corner of the ceiling. (c) The angle the space diagonal makes with the floor.
A vertical pole is 10 m tall. A wire is attached from the top to a point on level ground. The wire makes an angle of 65° with the ground. Find the length of the wire and the horizontal distance from the base to where the wire is anchored.
Pythagorean Triples
Draw a line from each set of three numbers to the statement that correctly describes it.
Bearings and Trigonometry
Solve navigation problems using bearings and trigonometry.
A ship sails 50 km on a bearing of 060° (N60°E). How far North and how far East of the starting point is the ship? Draw a diagram. Use sin and cos of 60°.
From a point P, town A is 80 km due North. Town B is 60 km due East of P. Calculate: (a) The straight-line distance from A to B. (b) The bearing of B from A (measured clockwise from North).
Steps to Solve a Trigonometry Problem
Put the steps in the correct order.
The Sine Rule — Introduction
Apply the sine rule to non-right triangles.
In triangle ABC: angle A = 40°, angle B = 75°, side a = 12 cm (opposite to A). Use the sine rule (a/sin A = b/sin B) to find side b. Then find angle C and side c.
Explain when you would use the sine rule rather than standard right-triangle trigonometry. What must be true about the triangle?
The Cosine Rule — Introduction
Apply the cosine rule to find missing sides and angles.
In triangle ABC: a = 8 cm, b = 11 cm, C = 60°. Use the cosine rule (c² = a² + b² − 2ab cos C) to find side c.
In triangle ABC: a = 5 cm, b = 7 cm, c = 9 cm. Use the cosine rule in rearranged form to find angle C (cos C = (a² + b² − c²) / 2ab).
Which Rule to Use?
Circle the most appropriate rule for each triangle problem.
Triangle with one right angle — find a side given an angle and another side
Triangle with two sides and the included angle — find the third side
Triangle with two angles and one side — find another side
Triangle with all three sides — find an angle
Trigonometry in Your Environment
Find and measure real-world angles using trigonometry.
- 1Use a protractor and ruler to measure the angle of elevation of the top of a building or tree. Measure the horizontal distance. Calculate the height using tan(angle).
- 2Look up a map of your town. Find two landmarks. Using a scaled map, measure the distance and calculate the bearing from one to the other.
- 3Stand at the base of a hill or ramp. Estimate the angle of the slope using a phone's inclinometer app. Measure the horizontal distance. Calculate the vertical height gained when walking up.
Trigonometry Applications — Field
Sort each application into its field.
Trigonometry Methods Used in a Problem Set
Record how often each method was used across 40 problems.
| Item | Tally | Total |
|---|---|---|
SOH CAH TOA (right triangle) | ||
Pythagoras' theorem | ||
Sine rule | ||
Cosine rule |
Area of a Non-Right Triangle
Apply the trigonometric area formula.
The area of any triangle can be found using: Area = ½ab sin C, where a and b are two sides and C is the included angle. For triangle with a = 9 cm, b = 14 cm, C = 70°: calculate the area. Verify by also finding the height using sin C and applying ½ × base × height.
Trig Ratios — Match to Definition
Draw a line from each trigonometric ratio to its definition.
Angles of Elevation and Depression
Solve practical problems involving angles of elevation and depression.
From the top of a 40 m cliff, the angle of depression to a boat is 25°. How far is the boat from the base of the cliff?
A person stands 80 m from a building and measures the angle of elevation to the top as 52°. Find the height of the building.
Explain the difference between angle of elevation and angle of depression with a diagram.
Exact Trigonometric Values
Circle the correct exact value.
sin 30°
cos 60°
tan 45°
sin 90°
cos 0°
Which Trig Rule to Use?
Sort each problem into which rule is most appropriate.
Bearings and Navigation
Apply trigonometry to navigation problems using bearings.
A ship travels 50 km on a bearing of 060°T, then 80 km on a bearing of 150°T. Draw a diagram and find the direct distance back to the start.
A plane flies 200 km due north then 120 km due east. Find the bearing of its final position from the starting point.
Explain how bearings are measured and why they always have three digits.
Trig Problem Types Practised
Tally each type of trigonometry problem you solved this week.
| Item | Tally | Total |
|---|---|---|
Right-angled trig | ||
Sine rule | ||
Cosine rule | ||
Bearing problems | ||
Area of triangle |
3D Trigonometry Problems
Apply trigonometry to three-dimensional scenarios.
A 5 m ladder leans against a wall, making a 72° angle with the ground. How high up the wall does it reach?
A rectangular room is 8 m long, 5 m wide and 3 m high. Find the length of the space diagonal (corner to corner).
A cone has base radius 6 cm and slant height 10 cm. Find the vertical height and the half-angle at the apex.
Trigonometry in Architecture and Nature
Find trigonometry in the world around you.
- 1Measure the angle of inclination of a ramp, staircase, or driveway at your home using a protractor or phone app. Calculate the horizontal and vertical distances.
- 2Research how ancient Egyptians used trigonometry to build the pyramids. Write a brief summary.
- 3Look at a satellite image of your neighbourhood. Estimate the angle of a road on a hill using rise over run.
- 4Use a clinometer (or phone app) to measure the height of a tall tree or building near your home. Show all calculations.
- 5Draw a triangle representing a roof pitch (e.g. 30° angle). Calculate what materials would be needed for a 10 m wide house.
Unit Circle Introduction
Begin exploring how trigonometry extends beyond right angles.
Draw a unit circle (radius 1) and mark the angles 0°, 90°, 180°, 270°, 360°.
For a point on the unit circle at angle θ, the coordinates are (cos θ, sin θ). Use this to find cos 90° and sin 180°.
In which quadrant is sin negative and cos positive? Explain.
Use the unit circle to explain why sin²θ + cos²θ = 1 for any angle.
Applying the Sine and Cosine Rules
Use the sine and cosine rules in non-right triangles.
In triangle ABC: a = 12, b = 9, C = 55°. Find c using the cosine rule.
In triangle PQR: p = 7, Q = 48°, R = 62°. Find q using the sine rule.
Two sides of a triangle are 8 cm and 11 cm, and the included angle is 130°. Find the third side.
Explain when the sine rule has two possible solutions (the ambiguous case). Give an example.
Trigonometry Problem Context
Sort each trigonometry problem into its real-world context.
Pythagorean Proofs
Explore different proofs of Pythagoras' theorem.
State Pythagoras' theorem and draw a right triangle with sides labelled a, b, and c.
Describe one proof of Pythagoras' theorem (e.g. using four copies of a right triangle arranged inside a square). Explain the key steps.
Verify Pythagoras' theorem for the Pythagorean triple (5, 12, 13). Calculate all three sides of a triangle with these proportions and check a² + b² = c².
Can Pythagoras' theorem be extended to 3D? Write the formula for the space diagonal of a rectangular box.