National 5 Maths: Trig Identities, the Sine Rule for an Angle and Bearings

Two parts of the National 5 Mathematics trigonometry content are often left until last in revision: the identities $\sin^{2} x^{\circ} + \cos^{2} x^{\circ} = 1$ and $\tan x^{\circ} = \frac{\sin x^{\circ}}{\cos x^{\circ}}$ , which appear as short simplify-or-prove questions on the non-calculator paper, and bearing problems, which combine three-figure bearings with the sine and cosine rules on the calculator paper. This material trains both, together with the sine rule used to find an angle.

The quiz has twelve questions. Four simplify expressions such as $3 - 3 \sin^{2} x^{\circ}$ and $\frac{\sin x^{\circ}}{\tan x^{\circ}}$ or use the identity to find $\cos x^{\circ}$ from $\sin x^{\circ}$ , and one asks for the right way to begin a proof. One uses the sine rule to find an angle and one the cosine rule to find the largest angle of a triangle of towns. The bearing questions cover back bearings, turning clockwise from a given bearing, the reason the angle at a turning point of a journey takes its value, and the distance travelled from the start, found with the cosine rule. A final question asks when the sine rule applies. All lengths and angles are stated in the text, and several triangle and journey problems come with a diagram, as on the exam, which shows the setup but never the value asked for; the wrong options come from typical slips such as subtracting instead of adding $180^{\circ}$ , confusing $\cos^{2} x^{\circ}$ with $\cos x^{\circ}$ , or using the turn instead of the angle inside the triangle.

The flashcards state the two identities and their rearrangements, what a three-figure bearing is, how to find a back bearing, when to choose the sine rule or the cosine rule for a side or an angle, why alternate and co-interior angles can be used between north lines, and how to set out the proof of an identity.

The printable written work has eight longer questions to answer by hand: two proofs, exact values of $\sin x^{\circ}$ and $\tan x^{\circ}$ from $\cos x^{\circ}$ , a sine rule angle problem, a two-leg journey where you find a distance and then a bearing, a back-bearing explanation, a ship located from two coastguard stations, and a reasoning question about impossible values of sine and cosine. Each asks for full working and a short explanation, as National 5 marking requires.

In the oral practice exam an examiner asks one question at a time, asks you to justify each step, and finishes with brief feedback. The material suits S4 pupils preparing for National 5 and anyone about to start Higher Mathematics.

Simplify expressions using sin² x + cos² x = 1 and tan x = sin x / cos x
Set out a short proof of a trigonometric identity
Use the sine rule and the cosine rule to find an angle in a triangle
Work with three-figure bearings and back bearings
Find the angle at a turning point from bearings, then a distance or direction with trigonometry

Practice material written by Zestly, based on the National 5 Mathematics course specification (Qualifications Scotland, formerly SQA; version 3.0, May 2023), trigonometric skills: identities, the sine and cosine rules for an angle, and bearings with trigonometry.

Sample question

A ship's journey: from the start, 12 km on a bearing of 050°, then from the turning point 8 km on a bearing of 130°, with north lines at the start and at the turning point; the angle at the turning point is marked 100° and the dashed line from the start to the end point is marked with a question mark.

A ship sails 12 km on a bearing of $050^{\circ}$ and then 8 km on a bearing of $130^{\circ}$. The angle at the turning point is $100^{\circ}$. Find the distance from the start to the end point to 1 decimal place.

15.5 km
14.4 km
16.2 km
13.8 km

See the answer

15.5 km

Using the cosine rule $b^2 = a^2 + c^2 - 2ac \cos(B)$, where $a=12$, $c=8$, and $B=100^{\circ}$: $b^2 = 12^2 + 8^2 - 2(12)(8) \cos(100^{\circ}) = 144 + 64 - 192(-0.1736) \approx 208 + 33.33 = 241.33$. $b = \sqrt{241.33} \approx 15.5$ km.

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National 5 and Higher qualifications are awarded by Qualifications Scotland, which took over the awarding functions of the Scottish Qualifications Authority (SQA) in February 2026. Zestly is not affiliated with or endorsed by Qualifications Scotland or the SQA. These materials are our own and reproduce no past-paper content.