Higher Maths: Harder Trigonometric Equations

Solving a trigonometric equation is one of the skills the Higher Mathematics course specification lists explicitly: in degrees or radians, including equations involving the wave function or trigonometric formulae or identities, in a given interval. The category's trigonometry material practises the formulae themselves and simple equations such as $2\cos x + 1 = 0$; this material moves on to the equations that carry the most marks and cause the most lost solutions.

It starts with multiple angles. For $\sin 2x=\frac{\sqrt{3}}{2}$ or $\cos 3x=0.5$ you first widen the interval for $2x$ or $3x$, find every solution there, and only then divide, which explains why $\sin 2x=\frac{1}{2}$ has four solutions in one turn of $x$.

It then turns to equations that become quadratics. Some are quadratic in $\sin x$ already; others need a double-angle formula first. You learn to choose the form of $\cos 2x$ that matches the rest of the equation, $1 - 2\sin^2 x$ next to a $\sin x$ term and $2\cos^2 x - 1$ next to a $\cos x$ term, to factorise, to reject a value of $\cos x$ outside $-1$ to $1$, and to find every angle in the interval, including end points when the interval includes them. Equations such as $\sin 2x=\sin x$ show why you factorise instead of dividing by $\sin x$.

The last group uses the wave function. You express $a\cos x+b\sin x$ as $k\cos (x-\alpha )$, $k\cos (x+\alpha )$ or $k\sin (x+\alpha )$ with the quadrant of $\alpha$ fixed by the signs, solve an equation by working over the shifted interval for $x\pm \alpha$, and find the maximum value of an expression such as $4 + 3\cos x - 4\sin x$ and where it occurs.

Some questions are non-calculator and rely on exact values, as in question paper 1; others give answers to one decimal place or in radians to two decimal places, as a calculator question would.

The material offers three formats. The quiz has 12 questions whose explanations list every solution and name the slip behind each wrong option, such as stopping after one turn or forgetting the negative square root. The flashcards collect the double-angle formulae, the rule for choosing the right form, the quadrant signs and the wave-function method. The printable written work has 8 longer equations to solve in full, including a quadratic with one impossible root, two wave-function problems in degrees and radians, and two explanations of where solutions get lost.

It is aimed at S5 and S6 students preparing for Higher Mathematics.

• Solve equations with multiple angles such as sin 2x = c over the full interval
• Solve trigonometric equations that are quadratic in sin x or cos x
• Choose and apply the correct double-angle formula to reduce an equation to one function
• Reject impossible values and include every solution in the interval, in degrees or radians
• Use the wave function to solve a cos x + b sin x = c
• Find the maximum or minimum of a wave function and where it occurs

Practice material written by Zestly, based on the Qualifications Scotland (formerly SQA) Higher Mathematics course specification, version 3.0 (algebraic and trigonometric skills: solving trigonometric equations in degrees or radians, including those involving the wave function or trigonometric formulae or identities, in a given interval).