Solving Trigonometric Equations Graphically
Not all trigonometric equations are easy to solve algebraically — especially when different trig functions are mixed together or when the equation involves transformations. In this lesson you will learn how to use graphs to find approximate solutions, count the number of solutions in a given interval, and verify algebraic answers by visual inspection.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Quick warm-up — sketch $y = \sin x$ and $y = 0.5$ on the same axes. How many intersections in $[0, 2\pi]$?
There are only two steps to solving any trigonometric equation graphically: draw both sides of the equation on the same axes, then read off the $x$-coordinates of every intersection point. The number of solutions in any interval equals the number of intersections in that interval.
Every graphical trig solution follows the same pattern: rewrite so one side is a single trig function and the other is a constant, sketch both graphs on the same axes, then read off the $x$-coordinates of intersection points. Count intersections to count solutions.
read $x$ at intersections
Key facts
- How to set up a graphical solution for trig equations
- That periodic functions can have infinitely many solutions
- How domain restrictions limit the number of solutions
Concepts
- Why the intersection of two graphs gives the solutions to an equation
- How symmetry helps locate all solutions in one period
- When graphical methods are more practical than algebraic methods
Skills
- Solve trig equations by sketching appropriate graphs
- Count the number of solutions in a given interval
- Verify algebraic solutions using graphical reasoning
To solve a trigonometric equation graphically, rewrite it so that one side is a trigonometric function and the other side is a constant or another function. Then sketch both graphs on the same axes and find their points of intersection.
Example: Solving $\sin x = 0.5$
Draw $y = \sin x$ and $y = 0.5$ on the same axes. In the interval $0 \leq x \leq 2\pi$, the horizontal line $y = 0.5$ cuts the sine curve twice: once in the first quadrant and once in the second quadrant. The solutions are $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$.
Example: Solving $\cos x = -0.5$
Draw $y = \cos x$ and $y = -0.5$. In $0 \leq x \leq 2\pi$, the line cuts the cosine curve twice: in the second and third quadrants. The solutions are $x = \frac{2\pi}{3}$ and $x = \frac{4\pi}{3}$.
To solve $\sin x = k$ graphically: sketch $y = \sin x$ and $y = k$ on the same axes; The solutions are the $x$-coordinates of every intersection point
Pause — copy the graphical method: sketch $y = \sin x$ and $y = k$ on the same axes; solutions are the $x$-coordinates of every intersection point into your book.
Quick check: How many solutions does $\sin x = 0.5$ have in $0 \leq x \leq 2\pi$?
We just saw that solutions to $\sin x = k$ are the $x$-coordinates where the horizontal line $y = k$ crosses the sine curve. That raises a question: for a given interval, how many intersections should we expect — and how does the period help us predict this? This card answers it → sine and cosine have period $2\pi$, so each full cycle contributes 2 solutions for most values of $k$.
The number of solutions to a trig equation in a given interval equals the number of intersections between the relevant graphs in that interval.
$\sin x = 0.3$ has 4 solutions in $[0, 4\pi]$ — two per period — shown where the sine curve intersects the horizontal line.
Example: How many solutions does $\sin x = 0.3$ have in $0 \leq x \leq 4\pi$?
The sine graph completes two full cycles in $4\pi$. The horizontal line $y = 0.3$ cuts each cycle twice. Therefore, there are $2 \times 2 = 4$ solutions.
Example: How many solutions does $\tan x = 1$ have in $0 \leq x < 2\pi$?
The tangent graph has period $\pi$, so there are two branches in $[0, 2\pi)$. Each branch intersects $y = 1$ exactly once. Therefore, there are 2 solutions: $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$.
Number of solutions in an interval = number of intersections in that interval; Sine and cosine have period $2\pi$ — each cycle contributes 2 intersections for most values of $k$
Pause — copy the counting rule: each period of $\sin/\cos$ contributes 2 intersections for most $k$; count intersections in the given interval directly from the sketch into your book.
True or false: $\sin x = 0.7$ has exactly 6 solutions in $[0, 6\pi]$.
Worked examples · 3 in a row, reveal as you go
By sketching $y = \sin x$ and $y = \cos x$ on the same axes, find all solutions to $\sin x = \cos x$ in $0 \leq x \leq 2\pi$.
How many solutions does $\sin x = 0.2$ have in $0 \leq x \leq 4\pi$?
By considering the graphs of $y = 2\sin x$ and $y = 1$, find all solutions to $2\sin x = 1$ in $0 \leq x \leq 2\pi$.
Common errors · the 3 traps that cost marks
Quick-fire practice · 5 reps +2 XP per reveal
$\cos x = 0.5$ for $0 \leq x \leq 2\pi$ — state all solutions.
$\sin x = -0.5$ for $0 \leq x \leq 2\pi$ — state all solutions.
$\tan x = \sqrt{3}$ for $0 \leq x \leq 2\pi$ — state all solutions.
How many solutions does $\sin x = 0.7$ have in $[0, 6\pi]$? Explain.
By sketching, find all solutions to $\sin x = \cos x$ in $[0, 2\pi]$.
Earlier you were asked about $\sin x = 0.5$ and infinitely many solutions.
In $0 \leq x \leq 2\pi$, the horizontal line $y = 0.5$ cuts the sine graph twice: at $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$. Because sine repeats every $2\pi$, the general solution is $x = \frac{\pi}{6} + 2\pi n$ or $x = \frac{5\pi}{6} + 2\pi n$ for any integer $n$. So there are infinitely many solutions overall, but only two in one full cycle. Domain restrictions are what turn infinite solutions into a finite, countable set.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. By sketching $y = \sin x$ and $y = \cos x$ on the same axes, find all solutions to $\sin x = \cos x$ in $0 \leq x \leq 2\pi$. (3 marks)
Q2. How many solutions does $\sin x = 0.7$ have in $0 \leq x \leq 6\pi$? Explain your reasoning. (2 marks)
Q3. The equation $\cos x = 0.5$ has two solutions in $[0, 2\pi]$. By considering the graph of $y = \cos x$, explain what happens to the number of solutions if the domain is extended to $[0, 4\pi]$. (3 marks)
Comprehensive answers (click to reveal)
Drill 1: $x = \frac{\pi}{3}, \frac{5\pi}{3}$
Drill 2: $x = \frac{7\pi}{6}, \frac{11\pi}{6}$
Drill 3: $x = \frac{\pi}{3}, \frac{4\pi}{3}$
Drill 4: 6 solutions (3 cycles × 2 intersections). Sine has period $2\pi$, so $6\pi$ contains 3 complete cycles. Each cycle intersects $y = 0.7$ twice.
Drill 5: $x = \frac{\pi}{4}, \frac{5\pi}{4}$. The graphs intersect where $\tan x = 1$.
Q1 (3 marks): Sketch both graphs [1]. Intersections occur where $\tan x = 1$ [1]. Solutions: $x = \frac{\pi}{4}, \frac{5\pi}{4}$ [1].
Q2 (2 marks): Sine has period $2\pi$, so $6\pi$ contains 3 cycles [1]. Each cycle intersects $y = 0.7$ twice, so there are 6 solutions [1].
Q3 (3 marks): In $[0, 2\pi]$, $y = 0.5$ cuts $y = \cos x$ twice [1]. Cosine has period $2\pi$, so in $[0, 4\pi]$ the pattern repeats [1]. The number of solutions doubles to 4 [1].
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