Solving Simple Trigonometric Equations
Theory
A trigonometric equation such as \(\sin x = 0.5\) usually has more than one solution, because the ratios repeat. This Year 10A maths page shows how to find the related angle, use the ASTC rule to place the answers, solve with exact values or a calculator, handle squares, smaller domains and two full turns, and read solutions from a graph. Curriculum: Year 10A (Victorian Curriculum 10A: VC2M10ASP04); Australian Curriculum: optional content beyond Year 10.
A trigonometric equation such as \(\sin x = 0.5\) asks for every angle in a given domain (such as \(0^\circ \le x \le 360^\circ\)) whose sine is \(0.5\). Because the ratios repeat, there is usually more than one answer.
The calculator's \(\sin^{-1}\), \(\cos^{-1}\) and \(\tan^{-1}\) give only one angle. Use it to find the related acute angle \(\alpha\), always from the positive value, then use the ASTC rule to decide which quadrants the answers are in.
Each ratio has the same size in four places, \(\alpha\), \(180^\circ - \alpha\), \(180^\circ + \alpha\) and \(360^\circ - \alpha\), but the right sign in only two of them. So \(\sin x = k\), \(\cos x = k\) and \(\tan x = k\) normally have two solutions from \(0^\circ\) to \(360^\circ\), and two more for each extra \(360^\circ\).
On a graph, the solutions are the \(x\)-values where the curve meets the horizontal line \(y = k\).
| Quadrant | Answer | Positive there |
|---|---|---|
| 1 | \(\alpha\) | all |
| 2 | \(180^\circ - \alpha\) | \(\sin\) |
| 3 | \(180^\circ + \alpha\) | \(\tan\) |
| 4 | \(360^\circ - \alpha\) | \(\cos\) |
| Equation | Quadrants of the answers |
|---|---|
| \(\sin x = k > 0\) | 1 and 2 |
| \(\sin x = k < 0\) | 3 and 4 |
| \(\cos x = k > 0\) | 1 and 4 |
| \(\cos x = k < 0\) | 2 and 3 |
| \(\tan x = k > 0\) | 1 and 3 |
| \(\tan x = k < 0\) | 2 and 4 |
Exact values for the related angle:
| \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | |
|---|---|---|---|
| \(\sin\) | \(\dfrac{1}{2}\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{\sqrt{3}}{2}\) |
| \(\cos\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{\sqrt{2}}\) | \(\dfrac{1}{2}\) |
| \(\tan\) | \(\dfrac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) |
How to solve a trig equation
- Isolate the ratio: \(3\cos x + 2 = 0\) gives \(\cos x = -\dfrac{2}{3}\). For a square such as \(\cos^2 x = k\), take both square roots, \(\pm\sqrt{k}\).
- Find the related angle \(\alpha\) from the positive value, by exact values or \(\sin^{-1}\), \(\cos^{-1}\), \(\tan^{-1}\).
- Choose the quadrants where the ratio has the right sign (ASTC), and place \(\alpha\) in each.
- List every answer in the domain, adding \(360^\circ\) if the domain is bigger, and round as asked.
- Check one answer on the calculator.
Other questions you will meet
- From a graph: draw the line \(y = k\) and read the \(x\)-values where it meets the curve. Count the crossings to see how many solutions there are.
- Models: a formula such as \(h = 5 - 3\sin x\) for a height: substitute the height, isolate \(\sin x\), then solve as usual.
- No solution: \(\sin x\) and \(\cos x\) are never bigger than \(1\) or less than \(-1\), so an equation such as \(\sin x = 1.4\) has no solution.
- Boundary values: \(\sin x = 1\), \(0\) or \(-1\) and the like have answers on the axes (\(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), \(360^\circ\)); read them from the graph.
(i) \(\sqrt{2}\sin\theta + 1 = 0\)
(ii) \(\tan\theta + 1 = 0\)
(i) Isolate the ratio:
| \(\sqrt{2}\sin\theta\) | \(=\) | \(-1\) |
| \(\sin\theta\) | \(=\) | \(-\dfrac{1}{\sqrt{2}}\) |
Related angle \(45^\circ\). Sine is negative in quadrants 3 and 4:
| \(\theta\) | \(=\) | \(180^\circ + 45^\circ = \mathbf{225^\circ}\) |
| \(\text{or } \theta\) | \(=\) | \(360^\circ - 45^\circ = \mathbf{315^\circ}\) |
(ii) \(\tan\theta = -1\), related angle \(45^\circ\). Tangent is negative in quadrants 2 and 4:
| \(\theta\) | \(=\) | \(180^\circ - 45^\circ = \mathbf{135^\circ}\) |
| \(\text{or } \theta\) | \(=\) | \(360^\circ - 45^\circ = \mathbf{315^\circ}\) |
(i) \(\cos x = -0.7\)
(ii) \(\tan x = -3\)
(i) Find the related angle from the positive value. Cosine is negative in quadrants 2 and 3:
| \(\alpha\) | \(=\) | \(\cos^{-1}(0.7) = 45.572\ldots^\circ\) |
| \(x\) | \(=\) | \(180^\circ - \alpha = 134.427\ldots^\circ\) |
| \(\text{or } x\) | \(=\) | \(180^\circ + \alpha = 225.572\ldots^\circ\) |
| \(x\) | \(\approx\) | \(\mathbf{134^\circ} \text{ or } \mathbf{226^\circ}\) |
(ii) Tangent is negative in quadrants 2 and 4:
| \(\alpha\) | \(=\) | \(\tan^{-1}(3) = 71.565\ldots^\circ\) |
| \(x\) | \(=\) | \(180^\circ - \alpha = 108.434\ldots^\circ\) |
| \(\text{or } x\) | \(=\) | \(360^\circ - \alpha = 288.434\ldots^\circ\) |
| \(x\) | \(\approx\) | \(\mathbf{108^\circ} \text{ or } \mathbf{288^\circ}\) |
Typing \(\cos^{-1}(-0.7)\) gives only \(134^\circ\); the related-angle method finds both.
(ii) Solve \(2\sin^2\theta = 1\) for \(0^\circ \le \theta \le 360^\circ\).
(i) Only quadrants 1 and 2 are in the domain, and sine is positive in both:
| \(\alpha\) | \(=\) | \(\sin^{-1}(0.35) = 20.487\ldots^\circ\) |
| \(\theta\) | \(\approx\) | \(\mathbf{20.5^\circ} \text{ or } 180^\circ - 20.487\ldots^\circ \approx \mathbf{159.5^\circ}\) |
(ii) Isolate \(\sin^2\theta\) and take both square roots:
| \(\sin^2\theta\) | \(=\) | \(\dfrac{1}{2}\) |
| \(\sin\theta\) | \(=\) | \(\dfrac{1}{\sqrt{2}} \text{ or } -\dfrac{1}{\sqrt{2}}\) |
Related angle \(45^\circ\) in every quadrant, one answer in each:
| \(\theta\) | \(=\) | \(\mathbf{45^\circ},\ \mathbf{135^\circ},\ \mathbf{225^\circ},\ \mathbf{315^\circ}\) |
The domain holds two full waves, and the line \(y = 0.8\) cuts each wave twice, so there are \(\mathbf{4}\) solutions.
| \(\alpha\) | \(=\) | \(\sin^{-1}(0.8) = 53.130\ldots^\circ\) |
| \(\text{first turn: } x\) | \(\approx\) | \(53.1^\circ,\ 180^\circ - 53.1^\circ = 126.9^\circ\) |
Add \(360^\circ\) to each for the second turn:
| \(x\) | \(\approx\) | \(\mathbf{53.1^\circ},\ \mathbf{126.9^\circ},\ \mathbf{413.1^\circ},\ \mathbf{486.9^\circ}\) |
Common pitfalls
Frequently asked questions
Why does a trig equation have two answers between 0 and 360 degrees?
Each ratio is positive in two quadrants and negative in the other two. So a value such as \(\sin x = 0.7\) is reached once in each of two quadrants, at about \(44.4^\circ\) and \(135.6^\circ\).
How do you solve sin x = a negative number?
Find the related angle from the positive value, for example \(\sin^{-1}(0.6)\). Sine is negative in quadrants 3 and 4, so the answers are \(180^\circ + \alpha\) and \(360^\circ - \alpha\).
Why does cos x = k have only one answer from 0 to 180 degrees?
Cosine is positive for every acute angle and negative for every obtuse angle, and it takes each value only once in that range. So \(\cos^{-1}\) gives the one answer directly.
How do you solve a trig equation with a square?
Make the squared ratio the subject, then take both the positive and the negative square root. Solve each one; there are usually four answers from \(0^\circ\) to \(360^\circ\).
How do you use a graph to solve a trig equation?
Draw the horizontal line \(y = k\) on the graph of the ratio. Each point where it meets the curve gives a solution, so counting the crossings tells you how many solutions there are.