National 5 Maths: Trigonometric Graphs, Related Angles and Equations

Trigonometric graphs and equations are one of the parts of National 5 Mathematics that pupils find hardest to revise from a textbook, because every question mixes reading a graph with knowing which quadrant an angle lies in. This material trains the trigonometric skills of the course that deal with the graphs of sine, cosine and tangent and with angles from $0^\circ$ to $360^\circ$.

The quiz has thirteen questions. The graph questions ask for the amplitude of $y=4\sin x^{\circ }$, the period of $y=\cos 3x^{\circ }$, the maximum and minimum of a graph moved up by a vertical translation, the values of $a$ and $b$ in $y=a\sin b{x}^{\circ }$ from a description of the curve, the effect of a phase angle, and the position of the maximum of $y=3\cos (x+45{)}^{\circ }$. Every graph is described in words, so nothing depends on a picture. Other questions check the period of the tangent graph and which functions repeat every $360^\circ$. The related-angle questions use the signs of sine, cosine and tangent in each quadrant to find exact values such as $\cos {240}^{\circ }$ and $\tan {135}^{\circ }$. Two equations, one with sine and one with tangent, ask for both solutions between $0^\circ$ and $360^\circ$ correct to one decimal place; the wrong options come from the usual mistakes of choosing the wrong quadrants or the wrong inverse function.

The flashcards give short, precise definitions of amplitude, period, vertical translation and phase angle, the quadrant rule often remembered as CAST, the related-angle formulae for the second, third and fourth quadrants, the exact values of sine, cosine and tangent for $30^\circ$, $45^\circ$ and $60^\circ$, and the periods of the three basic graphs.

The printable written work has eight longer questions to answer by hand: three equations with cosine, sine and tangent, describing and identifying graphs from their features, a set of exact values, a phase-angle graph, and a context problem about the height of a seat on a wheel, where you explain why an equation has two answers. Each asks for the working and a short explanation, as the exam does.

In the oral practice exam an examiner asks one question at a time, asks you to explain which quadrants you used and why, and ends with brief feedback. The material suits S4 pupils preparing for National 5 and is a useful refresher before the trigonometry of Higher.

• State the amplitude, period, maximum and minimum of graphs of the form y = a sin bx° + c and y = a cos bx° + c
• Describe the effect of a phase angle such as y = sin(x − 30)°
• Use related angles and the sign of each ratio in the four quadrants to find exact values
• Solve linear trigonometric equations for 0 ≤ x < 360 and give both solutions
• Relate a trigonometric equation to a context and explain its solutions

Practice material written by Zestly, based on the National 5 Mathematics course specification (Qualifications Scotland, formerly SQA; version 3.0, May 2023), trigonometric skills: graphs of trigonometric functions and trigonometric relationships in degrees.