Topic 6: Trigonometry

Topic 6 of Cambridge IGCSE Mathematics (0580) is one of the most heavily examined areas in Papers 2 and 4. It begins with right-angled triangles, where Pythagoras' theorem connects the three side lengths and the three trigonometric ratios connect sides with angles. It then extends to triangles of any shape through the sine and cosine rules, and into real-world contexts such as bearings, heights and distances, and three-dimensional figures. The topic closes with the behaviour of the trigonometric functions as graphs. Master the right-angled foundations first, because every later technique builds directly on them. Throughout, keep your calculator in degree mode and carry extra decimal places in working before rounding the final answer.

Pythagoras' Theorem

Pythagoras' theorem applies to right-angled triangles only. If the hypotenuse (the longest side, opposite the right angle) has length $c$ and the other two sides have lengths $a$ and $b$, then $a^2 + b^2 = c^2$. Picture a right-angled triangle with the right angle at the bottom-left corner; the slanted side opposite it is the hypotenuse. To find the hypotenuse, add the squares of the two shorter sides and take the square root: $c = \\sqrt{a^2 + b^2}$. To find a shorter side, subtract: $a = \\sqrt{c^2 - b^2}$. For example, with shorter sides $6$ and $8$, the hypotenuse is $\\sqrt{6^2 + 8^2} = \\sqrt{36 + 64} = \\sqrt{100} = 10$. A common error is subtracting when you should add: always check that the hypotenuse comes out as the largest side. Pythagoras also gives the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a coordinate grid: $d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

The Trigonometric Ratios: SOHCAHTOA

In a right-angled triangle, label the sides relative to a chosen angle $\\theta$: the hypotenuse (opposite the right angle), the opposite side (facing $\\theta$), and the adjacent side (next to $\\theta$). The three ratios are $\\sin\\theta = \\frac{O}{H}$, $\\cos\\theta = \\frac{A}{H}$ and $\\tan\\theta = \\frac{O}{A}$. The memory aid SOHCAHTOA stands for Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. To find a missing side, choose the ratio that uses the two sides you care about, substitute the known angle and side, then rearrange. For example, if $\\theta = 30^\\circ$ and the hypotenuse is $12$, the opposite side is $12 \\times \\sin 30^\\circ = 6$. To find a missing angle, use the inverse functions: $\\theta = \\sin^{-1}\\left(\\frac{O}{H}\\right)$, and similarly $\\cos^{-1}$ and $\\tan^{-1}$. For instance, if opposite $= 5$ and adjacent $= 7$, then $\\theta = \\tan^{-1}\\left(\\frac{5}{7}\\right) \\approx 35.5^\\circ$. Always identify which two sides are involved before picking the ratio.

Exact Values for Special Angles

Certain angles have exact trigonometric values worth memorising, because the exam may forbid a calculator or expect a surd answer. For $30^\\circ$: $\\sin 30^\\circ = \\frac{1}{2}$, $\\cos 30^\\circ = \\frac{\\sqrt{3}}{2}$, $\\tan 30^\\circ = \\frac{1}{\\sqrt{3}}$. For $45^\\circ$: $\\sin 45^\\circ = \\frac{1}{\\sqrt{2}}$, $\\cos 45^\\circ = \\frac{1}{\\sqrt{2}}$, $\\tan 45^\\circ = 1$. For $60^\\circ$: $\\sin 60^\\circ = \\frac{\\sqrt{3}}{2}$, $\\cos 60^\\circ = \\frac{1}{2}$, $\\tan 60^\\circ = \\sqrt{3}$. Also remember the boundary values $\\sin 0^\\circ = 0$, $\\cos 0^\\circ = 1$, $\\sin 90^\\circ = 1$ and $\\cos 90^\\circ = 0$. These come from two reference triangles: a right-angled isosceles triangle with two sides of $1$ and hypotenuse $\\sqrt{2}$ gives the $45^\\circ$ values, and half of an equilateral triangle of side $2$ gives the $30^\\circ$ and $60^\\circ$ values. Notice that $\\sin\\theta = \\cos(90^\\circ - \\theta)$, which is why the $30^\\circ$ and $60^\\circ$ values swap.

Angles of Elevation and Depression

These describe looking up or down from the horizontal. The angle of elevation is measured upward from a horizontal line to an object above, for instance the angle from the ground up to the top of a tower. The angle of depression is measured downward from a horizontal line to an object below, for instance from the top of a cliff down to a boat at sea. Imagine a horizontal dashed line from the observer's eye: the elevation angle opens above it, the depression angle opens below it. A key fact is that the angle of depression from a higher point equals the angle of elevation from the lower point back up, because they are alternate angles between parallel horizontal lines. To solve these problems, sketch a right-angled triangle with the horizontal distance, the vertical height and the line of sight, then apply SOHCAHTOA. For example, if a tower's top has an elevation of $40^\\circ$ from a point $50$ m away, the height is $50 \\times \\tan 40^\\circ \\approx 42.0$ m.

Bearings

A bearing is a direction measured clockwise from north, always written with three digits, so east is $090^\\circ$, south is $180^\\circ$ and west is $270^\\circ$. To find a bearing, draw a north arrow at the starting point, then measure the clockwise angle from that arrow round to the direction of travel. A bearing of $045^\\circ$ points north-east. Back bearings (the bearing of A from B compared with B from A) differ by $180^\\circ$: add $180^\\circ$ if the original is less than $180^\\circ$, otherwise subtract $180^\\circ$. Bearings questions usually combine with Pythagoras or the sine and cosine rules. Always draw a clear diagram with north lines at each relevant point, mark the known angles, and use the fact that the north lines are parallel so that co-interior angles between them sum to $180^\\circ$. For example, if a ship sails $060^\\circ$ for $8$ km then $150^\\circ$ for $6$ km, the two legs meet at a right angle, so the direct distance is $\\sqrt{8^2 + 6^2} = 10$ km.

The Sine Rule and Cosine Rule

For triangles that are not right-angled, label each angle with a capital letter and the side opposite it with the matching lowercase letter, so side $a$ is opposite angle $A$. The sine rule states $\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}$. Use it when you have a matching side-and-angle pair plus one more piece of information: either two angles and a side, or two sides and an angle opposite one of them. To find an angle, flip it: $\\frac{\\sin A}{a} = \\frac{\\sin B}{b}$. The cosine rule states $a^2 = b^2 + c^2 - 2bc\\cos A$. Use it when you know two sides and the included angle (to find the third side), or all three sides (to find an angle, by rearranging to $\\cos A = \\frac{b^2 + c^2 - a^2}{2bc}$). A quick decision guide: if a side and its opposite angle are paired, reach for the sine rule; otherwise use the cosine rule. For example, with sides $7$ and $9$ enclosing an angle of $60^\\circ$, the opposite side is $\\sqrt{7^2 + 9^2 - 2 \\times 7 \\times 9 \\times \\cos 60^\\circ} = \\sqrt{49 + 81 - 63} = \\sqrt{67} \\approx 8.19$.

Area of a Triangle and 3D Trigonometry

When you know two sides and the angle between them, the area of a triangle is $\\text{Area} = \\frac{1}{2}ab\\sin C$, where $C$ is the included angle between sides $a$ and $b$. For example, two sides of $10$ and $8$ enclosing $30^\\circ$ give area $\\frac{1}{2} \\times 10 \\times 8 \\times \\sin 30^\\circ = 20$ square units. In three-dimensional problems, such as finding an angle in a cuboid or a pyramid, the strategy is to identify a right-angled triangle inside the solid and work in two dimensions. Often you first use Pythagoras in three dimensions to find a diagonal, for instance the space diagonal of a cuboid with edges $a$, $b$ and $c$ is $\\sqrt{a^2 + b^2 + c^2}$. Then apply SOHCAHTOA to the right-angled triangle containing the required angle, frequently the angle between a sloping edge and the horizontal base. Drawing the relevant triangle separately, away from the cluttered 3D sketch, makes these much clearer.

Graphs of Sine, Cosine and Tangent

The graph of $y = \\sin x$ is a smooth wave starting at the origin, rising to a maximum of $1$ at $90^\\circ$, returning to $0$ at $180^\\circ$, falling to a minimum of $-1$ at $270^\\circ$, and back to $0$ at $360^\\circ$; it repeats every $360^\\circ$. The graph of $y = \\cos x$ has the same wave shape but starts at its maximum of $1$ when $x = 0^\\circ$, crosses zero at $90^\\circ$, reaches $-1$ at $180^\\circ$, and returns to $1$ at $360^\\circ$. Both oscillate between $-1$ and $1$ with a period of $360^\\circ$, and the cosine curve is the sine curve shifted left by $90^\\circ$. The graph of $y = \\tan x$ is very different: it rises steeply, repeats every $180^\\circ$, passes through zero at $0^\\circ$, $180^\\circ$ and $360^\\circ$, and has vertical asymptotes at $90^\\circ$ and $270^\\circ$ where the value shoots toward infinity and is undefined. These graphs explain why an equation such as $\\sin x = 0.5$ has more than one solution in the range $0^\\circ$ to $360^\\circ$, here $x = 30^\\circ$ and $x = 150^\\circ$.