0580 · 6.2Right-angled triangles

Trigonometric ratios in right-angled triangles

0:00A ramp’s length does not tell us how high it reaches. Its slope matters too, because a steeper ramp gains more height over the same length. Trigonometry connects that slope angle to the distances in a right-angled triangle. By the end, you will understand how to calculate heights, distances and angles without measuring them directly. We need three starting ideas: right angles, similar triangles and simple rearranging. A right angle is ninety degrees, and an acute angle lies between zero degrees and ninety degrees.

0:27Similar triangles have equal corresponding angles and proportional corresponding sides. Rearranging an equation lets us isolate the length or angle we want to calculate. Pythagoras’ theorem connects the three side lengths of a right-angled triangle. The square of the hypotenuse equals the sum of the squares of the other two sides. We covered that theorem in the previous video, so here we will use it rather than prove it again.

0:48Trigonometry adds a connection between side lengths and angles. The hypotenuse is the side opposite the right angle, and it is always the longest side. Choose one acute angle as your reference angle. The opposite side faces that angle across the triangle. The adjacent side touches that angle, but it is not the hypotenuse. These labels depend on the chosen angle, not on the page orientation. Rotating a triangle does not change which side is its hypotenuse.

1:13However, choosing the other acute angle swaps the opposite and adjacent sides. Always mark your reference angle before labelling those two sides. Imagine enlarging a triangle until every side is twice as long. The angles stay unchanged because the shape stays the same. An opposite side of three and a hypotenuse of five become six and ten. Both triangles therefore have the same opposite divided by hypotenuse ratio, namely three fifths.

1:38This explains why trigonometric ratios depend on the angle rather than the triangle’s size. Every right-angled triangle with the same chosen acute angle has the same side proportions. A calculator can give us those proportions for a particular angle. We can then use one known length to calculate another. The sine of an angle equals opposite divided by hypotenuse. Cosine compares adjacent with hypotenuse, while tangent compares opposite with adjacent.

2:04Each ratio divides one length by another measured in the same units. The units cancel, so the ratio itself has no length unit. The memory aid on the board, built from the first letters of each ratio, summarises these three relationships. It is useful, but the important idea is choosing the ratio containing the sides you need. If your problem involves opposite and hypotenuse, use sine. Adjacent and hypotenuse require cosine, while opposite and adjacent require tangent.

2:30As an acute angle increases, its opposite side becomes larger relative to its adjacent side. That means its tangent increases, matching our intuition about a steeper slope. Also, sine and cosine must lie between zero and one for acute angles. Each divides a shorter side by the hypotenuse, so neither can exceed one.

Calculator setup and ramp calculations

2:48Before using your calculator, check that it is in degree mode. A different angle mode gives different results, even when your triangle setup is correct. Give calculated angles to one decimal place unless instructed otherwise. Lengths usually need three significant figures unless the question gives another accuracy instruction. Our first exam-style example shows a ramp eight metres long, making an angle of thirty-two degrees with horizontal ground. We must calculate its height and horizontal distance.

3:15The ramp is the hypotenuse because it lies opposite the right angle. Relative to thirty-two degrees, the height is opposite and the horizontal distance is adjacent. The height and hypotenuse belong in the sine ratio, so sine of thirty-two degrees equals h ÷ 8. Multiplying both sides by eight gives h = 8 multiplied by sine of thirty-two degrees. The calculator gives four point two three nine metres, followed by more digits.

3:40To three significant figures, the height is four point two four metres. For the horizontal distance, cosine of thirty-two degrees equals d ÷ 8. Rearranging gives d = 8 multiplied by cosine of thirty-two degrees. This produces six point seven eight four metres, followed by more digits. The horizontal distance is therefore six point seven eight metres to three significant figures. Both answers are shorter than eight metres, which makes sense because neither is the hypotenuse.

4:09The height is also smaller than the horizontal distance, matching the fairly shallow slope. These checks cannot replace working, but they help identify mistakes. An impossible length often signals an incorrect ratio or calculator setting.

Finding unknown sides and angles

4:22Do not assume every missing side requires multiplication. If the hypotenuse is unknown, sine of the angle equals the known opposite side divided by that unknown length. Multiply through by the hypotenuse, then divide by sine of the angle. The hypotenuse therefore equals the opposite side divided by sine of the angle. Our second example asks for an angle rather than a length. Triangle PQR has a horizontal side of ten centimetres and a vertical side of seven centimetres.

4:47We want angle QPR, at P. The known sides are adjacent and opposite, so tangent connects them to the angle. Tangent of the angle equals 7 ÷ 10. Dividing gives a ratio of zero point seven. Apply the inverse tangent of zero point seven to find the angle. The result is thirty-four point nine nine two degrees, followed by more digits. Rounded to one decimal place, the angle is thirty-five point zero degrees.

5:12Inverse tangent reverses the original process by finding the angle that produces a given tangent ratio. It does not mean one divided by tangent. Inverse sine and inverse cosine work in the same way for their corresponding ratios. Choose the inverse function that matches your original equation.

Elevation, depression and tower heights

5:29An angle of elevation is measured upwards from a horizontal line to the line of sight. For an angle of depression, we measure downwards from the horizontal instead. Neither is measured from a vertical wall. Drawing the observer’s horizontal line is therefore an essential first step. When two observers look along the same line of sight, their elevation and depression angles are equal. Their horizontal lines are parallel, and the sight line creates equal alternate angles.

5:54This lets us transfer a depression angle into a right-angled triangle below it. We still need to identify the correct vertical height difference. In our third example, an observer stands twenty-four metres horizontally from a vertical tower. Their eye is one point six metres above level ground. The angle of depression from the tower’s top to their eye is thirty-five degrees. We must calculate the tower’s height and the line-of-sight distance.

6:16The angle of elevation from the observer’s eye is also thirty-five degrees because the horizontal lines are parallel. The triangle’s adjacent side is twenty-four metres. Its opposite side is the height above eye level, which we call y. The tower’s total height must include the additional one point six metres below that level. Tangent of thirty-five degrees equals y ÷ 24. Multiplying gives y = 24 multiplied by tangent of thirty-five degrees.

6:43This is sixteen point eight zero four metres, followed by more digits. Adding the eye height gives eighteen point four zero four metres, again followed by more digits. The tower’s height is eighteen point four metres to three significant figures. For the sight distance, Pythagoras gives the hypotenuse squared as 24² plus the unrounded height difference squared. These squares are five hundred and seventy-six and approximately two hundred and eighty-two point four zero seven.

7:10Adding gives approximately eight hundred and fifty-eight point four zero seven. Taking the square root gives twenty-nine point two nine eight metres, followed by more digits. The sight distance is twenty-nine point three metres to three significant figures. Keep the full calculator value of the height difference during this calculation. Rounding halfway through can change the final answer, especially in a longer problem. Also, use the height above eye level, not the total tower height, in this triangle.

7:37Its lower horizontal side passes through the observer’s eye rather than along the ground.

Shortest distances, bearings and exam technique

7:42The perpendicular distance from a point to a line is the shortest distance to the line. Perpendicular means meeting at ninety degrees. Any sloping route from the point to another position on the line forms a right-angled triangle. That route is the hypotenuse, so it must be longer than the perpendicular side. Pythagoras explains this precisely: the sloping distance squared equals the perpendicular distance squared plus an extra positive square.

8:05Moving the endpoint away from the perpendicular foot therefore increases the distance. A shortest-distance question is really asking you to construct a perpendicular. The diagram may not show that useful triangle until you draw it. Our fourth example places a boat five hundred metres from a harbour on a straight north to south shoreline. Its bearing from the harbour is zero three eight degrees. Bearings are measured clockwise from north, so the angle between the northward shoreline and the boat’s direction is thirty-eight degrees.

8:33We need the shortest distance from the boat to the shoreline. Draw a perpendicular from the boat to the shoreline to create the required right-angled triangle. Sine of thirty-eight degrees equals d ÷ 500. Rearranging gives d = 500 multiplied by sine of thirty-eight degrees. The calculator gives three hundred and seven point eight three zero metres, followed by more digits. The shortest distance is three hundred and eight metres to three significant figures.

8:58On Paper Two, the non-calculator paper, you must identify sides, select ratios, rearrange equations and recognise perpendicular distances without calculator support. Straightforward Pythagoras calculations must also work by hand. Decimal trigonometric evaluations like our examples use a calculator on Paper Four. Understanding the setup matters on both papers, even before numerical evaluation. Common exam mistakes include labelling sides before choosing the reference angle and treating a depression angle as an angle from the vertical.

9:29Avoid these by marking the angle and drawing the horizontal first. Another mistake is using an ordinary trigonometric function when finding an angle. Once you have a side ratio, use the matching inverse function. For an exam solution, show the ratio equation before entering numbers into your calculator. Each line should establish one clear step, ending with the requested quantity and unit. Retain unrounded values during intermediate calculations. Check that a hypotenuse is longest, and write calculated angles in degrees to one decimal place.

9:59The central idea is that equal angles create equal side ratios in similar right-angled triangles. Sine compares opposite with hypotenuse, cosine compares adjacent with hypotenuse, and tangent compares opposite with adjacent. Inverse functions recover angles from those ratios. Pythagoras connects side lengths when angles are not needed. For practical problems, the hardest step is often drawing the correct triangle. Elevation and depression begin at horizontal lines, while shortest distances meet lines perpendicularly.

10:28Bearings provide directions measured clockwise from north. Once those relationships are clear, the calculations become consequences of the diagram rather than formulas chosen by guesswork.