Right-triangle trigonometry is a relationship system, not three buttons on a calculator. Pythagoras links side lengths. Sine, cosine and tangent link an acute angle to side ratios. In three dimensions, the same ideas still work once the correct right triangle has been identified inside the solid.
This twenty-seventh Secondary 4 Mathematics Learning Guide develops Pythagoras, right-triangle trigonometry, elevation and depression, bearings and three-dimensional reasoning as one connected geometry system. It belongs to the Secondary Mathematics Hub and S1–S4 Capability Map.
Current syllabus connection: the 2026 O-Level Mathematics 4052 and 2027 SEC G3 Mathematics K310 syllabuses include Pythagoras, trigonometric ratios, elevation and depression, bearings, and solving two- and three-dimensional problems.
Pythagoras belongs only to right triangles
For a right triangle with perpendicular sides a and b and hypotenuse c:
a²+b²=c².
The hypotenuse is the side opposite the right angle and is always the longest side. If the triangle is not right-angled, this relationship does not apply directly.
Worked Example 1 | Missing hypotenuse
A right triangle has perpendicular sides 9 cm and 12 cm. Find the hypotenuse.
c²=9²+12²=81+144=225.
c=15 cm.
Check: 15 is longer than both perpendicular sides.
Worked Example 2 | Missing shorter side
A right triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other side x.
x²+5²=13², so x²=169−25=144.
x=12 cm.
A length is non-negative, so the geometric answer is 12 cm rather than ±12 cm.
SOHCAHTOA is a side-role map
Relative to an acute angle θ in a right triangle:
- sin θ = opposite/hypotenuse;
- cos θ = adjacent/hypotenuse;
- tan θ = opposite/adjacent.
The words opposite and adjacent depend on which acute angle is being used. The hypotenuse does not change.
Worked Example 3 | Find a side with sine
A right triangle has hypotenuse 18 cm and an acute angle of 37°. Find the side opposite the angle.
sin37°=x/18, so x=18sin37°≈10.83 cm.
The answer is less than the hypotenuse, as required.
Worked Example 4 | Find an angle with tangent
A right triangle has opposite side 7 cm and adjacent side 11 cm. Find θ.
tanθ=7/11, so θ=tan⁻¹(7/11)≈32.47°.
Check that the calculator is in degree mode when the answer is required in degrees.
Choose the ratio from the sides you know and need
If the known and unknown sides are opposite and adjacent, tangent is direct. If they are opposite and hypotenuse, sine is direct. If they are adjacent and hypotenuse, cosine is direct.
Pythagoras may be shorter when two sides are already known and the third side is required. Method choice should reduce unnecessary steps.
Elevation and depression are measured from horizontal lines
An angle of elevation is measured upward from a horizontal line. An angle of depression is measured downward from a horizontal line.
Horizontal lines at different heights are parallel. This often creates equal alternate angles between an observer’s horizontal line and the ground-level horizontal line.
Worked Example 5 | Height from angle of elevation
A point on level ground is 50 m from the base of a tower. The angle of elevation to the top is 28°. Ignore observer height. Find the tower height.
tan28°=h/50, so h=50tan28°≈26.59 m.
The separate Bearings, Sine Rule, Cosine Rule and Triangle Area guide handles non-right triangles in more depth.
Worked Example 6 | Angle of depression
From the top of a 36 m building, the angle of depression to a point on level ground is 41°. Find the horizontal distance d from the building.
The angle of elevation from the ground point to the top is also 41°.
tan41°=36/d, so d=36/tan41°≈41.41 m.
Three dimensions: find a hidden right triangle
A cuboid may require two Pythagoras steps. First find a diagonal on one rectangular face. Then combine that face diagonal with the third perpendicular dimension.
Worked Example 7 | Space diagonal of a cuboid
A cuboid measures 6 cm by 8 cm by 24 cm. Find the length of the space diagonal joining opposite vertices.
Base diagonal=√(6²+8²)=10 cm.
Now use the right triangle formed by base diagonal 10 cm and height 24 cm:
d=√(10²+24²)=√676=26 cm.
Equivalently, d=√(6²+8²+24²). The two-stage construction explains why that compact formula works.
Worked Example 8 | Angle between a space diagonal and a base
For the same 6×8×24 cuboid, find the angle θ between the 26 cm space diagonal and its 10 cm projection on the base.
In the right triangle, opposite=24 and adjacent=10:
tanθ=24/10, so θ≈67.38°.
The phrase “angle between a line and a plane” is interpreted using the line’s projection onto the plane.
Bearings can create right triangles
If one journey is due east and another is due north, the paths are perpendicular. Distance from the starting point can therefore be found using Pythagoras.
Worked Example 9 | East-north displacement
A boat travels 12 km due east and then 5 km due north. Find its straight-line distance from the starting point.
d=√(12²+5²)=13 km.
If the bearing from the start is also required, use the same right triangle to find the angle east of north or north of east, then convert to a three-figure bearing.
Worked Example 10 | Bearing from rectangular displacement
Using the 12 km east and 5 km north displacement, find the bearing of the boat from the starting point.
The bearing angle is measured clockwise from north. Let θ be the angle east of north:
tanθ=12/5, so θ≈67.38°.
Therefore the bearing is approximately 067° to the nearest degree.
Three-dimensional problems may need a diagram redraw
Perspective drawings can make right angles and relevant triangles hard to see. Redraw only the triangle you need, label its known sides, and mark the right angle explicitly.
This reduces visual load and prevents using a length that does not lie in the same plane as the required angle.
Worked Example 11 | Pyramid-style height from slant information
A vertical mast stands at the centre of a square platform. The distance from the centre to one corner is 6 m, and the cable from the top of the mast to that corner is 10 m. Find the mast height.
The mast, centre-to-corner distance and cable form a right triangle:
h²+6²=10², so h²=64 and h=8 m.
Common failure modes
| Error | Cause | Repair |
|---|---|---|
| Uses Pythagoras in a non-right triangle | Formula selected before angle structure | Confirm the 90° condition first |
| Calls the longest drawn-looking side the hypotenuse | Perspective trusted over geometry | Hypotenuse is opposite the right angle |
| Opposite and adjacent sides swapped | Reference angle not fixed | Mark θ before naming side roles |
| Angle of depression measured from vertical | Reference line forgotten | Draw a horizontal line through the observer |
| Uses a 3D length not in the target triangle | Solid not reduced to a plane | Redraw the relevant right triangle |
| Calculator in radian mode | Machine state not checked | Confirm degree mode for degree questions |
Independent practice
- A right triangle has perpendicular sides 7 cm and 24 cm. Find the hypotenuse.
- A right triangle has hypotenuse 20 cm and adjacent side 16 cm. Find the opposite side.
- A right triangle has opposite side 9 cm and hypotenuse 15 cm. Find the acute angle opposite the 9 cm side.
- A point is 30 m from a tower. The angle of elevation is 35°. Find the tower height, ignoring observer height.
- A cuboid measures 3 cm by 4 cm by 12 cm. Find its space diagonal.
- A person walks 8 km east and 6 km north. Find straight-line displacement and the bearing from the start.
Explained answers
1. √(7²+24²)=√625=25 cm.
2. √(20²−16²)=√144=12 cm.
3. sinθ=9/15=0.6, so θ≈36.87°.
4. h=30tan35°≈21.01 m.
5. Base diagonal=5; space diagonal=√(5²+12²)=13 cm.
6. Distance=√(8²+6²)=10 km. Bearing angle east of north satisfies tanθ=8/6, so θ≈53.13° and bearing≈053°.
Final thought
Right-triangle trigonometry becomes stable when the learner identifies the right triangle before choosing the formula. In three dimensions, the main challenge is often not calculation but locating the two-dimensional triangle hidden inside the solid.
Find the right angle. Fix the reference angle. Then let the side relationships do the work.
Return to the Secondary Mathematics Hub.
