Right-angled triangles
Section: Trigonometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Labelling a Right-Angled Triangle
Before using trigonometry, each side of a right-angled triangle must be correctly identified relative to a chosen angle.
- The hypotenuse is always the longest side, opposite the right angle - it never changes, regardless of which angle is chosen
- The opposite side is the side directly across from the chosen angle
- The adjacent side is the side next to the chosen angle (that isn't the hypotenuse)
- Which side is "opposite" and which is "adjacent" depends on which angle is being used - they swap if a different angle in the same triangle is chosen
Worked Example: Labelling the Sides of a Triangle
- Question: Triangle PQR has a right angle at Q. Angle R is marked. Identify the hypotenuse, the side opposite angle R, and the side adjacent to angle R.
- Step 1: The hypotenuse is opposite the right angle, so it is side PR
- Step 2: The side opposite angle R is side PQ
- Step 3: The side adjacent to angle R (that isn't the hypotenuse) is side QR
- Answer: hypotenuse = PR, opposite = PQ, adjacent = QR
Common Mistakes
MistakeAssuming the hypotenuse changes depending on which angle is chosen
Fixthe hypotenuse is always the same side - it is fixed by the position of the right angle, not by which other angle is being used
MistakeMixing up which side is opposite and which is adjacent when relabelling for a different angle
Fixrelabel the triangle freshly for each angle used - the side that was "adjacent" for one angle often becomes "opposite" for the other
Finding a Side Using Trigonometry
The three trigonometric ratios - sine, cosine and tangent - connect an angle in a right-angled triangle to the ratio of two of its sides.
- sin(angle)=(opposite/hypotenuse)
- cos(angle)=(adjacent/hypotenuse)
- tan(angle)=(opposite/adjacent)
- These are often remembered using "SOH CAH TOA". To find a missing side, choose the ratio that connects the known angle, the known side, and the unknown side
Worked Example: Finding a Side Using Sine
- Question: In a right-angled triangle, the hypotenuse is 18 cm, and one angle is 35°. Find the length of the side opposite this angle.
- Step 1: sin(35)=(opposite/18)
- Step 2: opposite=18(35)
- Step 3: 10.3
- Answer: 10.3 cm (3 s.f.)
Worked Example: Finding a Side Using Tangent
- Question: In a right-angled triangle, the side adjacent to a 42° angle is 14 cm. Find the length of the side opposite this angle.
- Step 1: tan(42)=(opposite/14)
- Step 2: opposite=14(42)
- Step 3: 12.6
- Answer: 12.6 cm (3 s.f.)
Common Mistakes
MistakeChoosing the wrong ratio (sin, cos or tan) for the two sides actually involved
Fixidentify which two sides are known/unknown first (hypotenuse, opposite, adjacent), then pick the ratio that matches that exact pair
MistakeDividing by the trig ratio when multiplying is needed, or vice versa
Fixrearrange the ratio equation carefully before calculating - if the unknown is on the top of the fraction, multiply; if on the bottom, divide
Finding an Angle Using Trigonometry
When two sides of a right-angled triangle are known, the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) are used to find an unknown angle.
- Find the ratio of the two known sides first, matching it to sin, cos or tan based on which sides they are
- Apply the inverse function (sin⁻¹, cos⁻¹ or tan⁻¹) to that ratio to find the angle
- The inverse trigonometric buttons on a calculator are usually labelled sin⁻¹, cos⁻¹ and tan⁻¹ (often accessed via a "shift" or "2nd function" key)
Worked Example: Finding an Angle Using Inverse Tangent
- Question: A right-angled triangle has an opposite side of 8.4 cm and an adjacent side of 6.1 cm, relative to angle x. Find x.
- Step 1: tan(x)=(8.4/6.1)
- Step 2: x=tan^-1((8.4/6.1))
- Step 3: 54.0
- Answer: x ≈ 54.0° (3 s.f.)
Common Mistakes
MistakeForgetting to use the inverse trig function, and instead just evaluating tan of the ratio (or another wrong operation)
Fixwhen the angle itself is unknown (and two sides are known), the inverse function (sin⁻¹, cos⁻¹, tan⁻¹) must be used, not the ordinary sin, cos or tan
MistakeMixing up which side is opposite and which is adjacent, leading to the reciprocal ratio and a wrong angle
Fixrelabel the triangle for the specific angle being found before writing the ratio
Applying Trigonometry to Angles of Elevation and Depression
Trigonometry is often applied to real-world situations involving looking up or down at an angle, such as the angle of elevation to the top of a building.
- The angle of elevation is the angle measured upward from the horizontal to a higher point
- The angle of depression is the angle measured downward from the horizontal to a lower point
- The angle of elevation from one point to another equals the angle of depression in the opposite direction (they are alternate angles between parallel horizontal lines)
- Sketch a right-angled triangle from the description first, labelling the known side, the known angle, and what needs to be found
Worked Example: Using an Angle of Elevation
- Question: From a point on the ground 25 m from the base of a tower, the angle of elevation to the top of the tower is 38°. Find the height of the tower.
- Step 1: The horizontal distance (25 m) is adjacent to the 38° angle; the height is opposite
- Step 2: tan(38)=(height/25)
- Step 3: height=25(38)19.5
- Answer: 19.5 m (3 s.f.)
Common Mistakes
MistakeConfusing the angle of elevation with the angle of depression, or measuring from the wrong horizontal line
Fixelevation is always measured looking up from the lower point; depression is always measured looking down from the higher point
MistakeNot sketching a triangle at all, leading to sides being mismatched or a wrong ratio being chosen
Fixalways sketch a quick right-angled triangle first, marking the known angle, known side and the horizontal/vertical relationship clearly
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