Right-angled triangles.
Choose the right ratio, locate the horizontal and calculate sides and angles in two dimensions.
Use sine, cosine and tangent · solve side and angle problems · identify shortest distance · use elevation, depression and bearings
Name the sides relative to the chosen angle
For a chosen acute angle in a right triangle, the opposite side faces that angle and the adjacent side touches it without being the hypotenuse. These labels depend on which acute angle you choose.
Similar right triangles with the same acute angle have the same ratios of corresponding sides. Sine, cosine and tangent name three of those fixed ratios. This explains why knowing an angle can link two side lengths even when the triangle is enlarged.
The hypotenuse is opposite the right angle. For your chosen acute angle θ, the opposite side is across from it; the adjacent side touches it and is not the hypotenuse.
θ = 35°; O ≈ 5.736; A ≈ 8.192.
sin θ = OH ≈ 0.5736
cos θ = AH ≈ 0.8192
tan θ = OA ≈ 0.7002
Pause and explainIf you switch to the other acute angle, do opposite and adjacent keep their labels?
No. They swap relative roles. The hypotenuse remains the side opposite the right angle.
Choose the ratio containing the known and unknown sides
Choose a trigonometric ratio by the information you have, not by the order of the mnemonic. Label the given side and the required side relative to the known angle.
Select the ratio containing those two sides, substitute carefully and rearrange the equation. If the angle is unknown, first calculate the relevant numerical ratio, then use the inverse trigonometric function. Ensure the calculator is in degree mode and round the final angle as requested.
A 6 m ladder makes an angle of 68° with level ground. Find the height reached.
Height is opposite the 68° angle and 6 m is the hypotenuse.
h = 6 sin 68° ≈ 5.56 m
An acute angle has opposite side 6 cm and adjacent side 8 cm. Find the angle.
θ = tan−1(68) ≈ 36.9°
Use the inverse trig key to recover an angle. sin−1 here means inverse sine, not a reciprocal.
The adjacent side is 7 cm and the angle is 40°. Find the hypotenuse.
H = 7cos 40° ≈ 9.14 cm
Pause and explainWhy use an inverse sine function when the sine ratio is known but the angle is not?
Sine maps an angle to a ratio. Inverse sine reverses that relationship to recover an appropriate angle.
The perpendicular is the shortest distance
There are many segments from a point to a line, but one is shortest: the perpendicular segment. Its right angle makes it useful for both geometry and calculation.
When a question asks for distance from a point to a line, draw that perpendicular rather than choosing any convenient sloping segment. It often creates a right triangle with a known angle or side, allowing Pythagoras or trigonometry to find the required distance.
Triangle PHQ is right-angled, so PQ is its hypotenuse and is longer than PH whenever Q ≠ H. A question asking for distance to a line means the perpendicular distance.
P is 30 m perpendicularly from a straight road. Q is 40 m along the road from the foot of that perpendicular. Find PQ.
The shortest distance from P to the road is still 30 m, not 50 m.
Pause and explainDoes the length of any line segment from the point to the line give the required perpendicular distance?
No. The required distance is the shortest segment, meeting the line at 90°.
Elevation, depression and bearings
An angle of elevation is measured upward from a horizontal line; an angle of depression is measured downward from a horizontal line. The horizontal reference belongs at the observer’s position.
Sketch the horizontal before marking the angle. If the object is above a building or person, include the observer’s height as well as the triangle’s vertical gap. For bearings, first identify the horizontal-plane direction, then isolate the right triangle needed for the calculation.
An angle of elevation looks upwards from the horizontal. An angle of depression looks downwards from the horizontal. In this diagram they are equal by alternate angles in parallel lines.
A tower is 45 m tall. From level ground 60 m from its base, find the angle of elevation of its top.
θ ≈ 36.9°
An observer’s eyes are 1.6 m above the ground, 30 m horizontally from a building. The angle of elevation to the roof is 28°.
The right-triangle height starts at eye level.
B is 12 km from A on bearing 060°. Find its eastward and northward displacements.
Draw north at A. The 60° angle is measured from north, so east is opposite and north is adjacent.
North = 12 cos 60° = 6 km
Pause and explainWhy is an elevation angle not normally measured from a vertical wall?
Its definition uses the observer’s horizontal line. An angle to the vertical would be a different, complementary angle.
Your turn. Draw the right triangle.
Label hypotenuse, opposite and adjacent relative to the angle used in that question. Choose a ratio only after those labels are clear.
Keep full precision while calculating and give final angles in degrees to the requested accuracy. In elevation and depression questions, distinguish the vertical gap from the full object height.
A right triangle has hypotenuse 12 cm and an acute angle 35°. Find the opposite side.
Reveal worked answer
Find the acute angle with opposite side 6 and adjacent side 8.
Reveal worked answer
From level ground 40 m from a tower, the angle of elevation is 28°. Find the tower height.
Reveal worked answer
A boat is 52 m horizontally from the foot of a 30 m cliff. Find the angle of depression from the cliff top.
Reveal worked answer
A point is 8 cm from a line along its perpendicular. Is a sloping segment of length 10 cm to the line the shortest distance?