Module 2 · L13 of 15~35 min⚡ +50 XP in Learn · +25 to complete
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.
Today's hook, Consider the equation $\sin x = 0.5$. You know one solution is $x = \frac{\pi}{6}$. But because the sine graph repeats forever, there must be infinitely many solutions. How would you find all of them? And if you restricted the domain to $0 \leq x \leq 2\pi$, how many solutions would there be?
0/5QUESTS
1
You’re here
Orient and recall
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
Worksheets
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 $[0, 2\pi]$?
auto-saved
2
You’re here
The two moves
Work through the core explanation before applying it.
02
The two moves
+5 XP to read
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.
$\sin x = k$ read $x$ at intersections
Graphical solution principle
To solve $\sin x = k$, draw $y = \sin x$ and $y = k$ on the same axes. The $x$-coordinates of intersection points are the solutions.
Intersections are solutions
At every intersection point, both graphs have the same $y$-value for the same $x$-value. That $x$-value satisfies the original equation.
Counting key insight
The number of solutions in a given interval equals the number of intersections between the relevant graphs in that interval.
3
You’re here
What you'll master
Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.
03
What you'll master
Know
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
Understand
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
Can do
Skills
Solve trig equations by sketching appropriate graphs
Count the number of solutions in a given interval
Verify algebraic solutions using graphical reasoning
04
Key terms
Trigonometric equation
An equation where the unknown sits inside sine, cosine or tangent. Like this: $\sin x = 0.5$ asks which angles have a sine of 0.5.
Restricted domain
The interval you are told to search, so the answer list is finite. Like this: "for $0 \leq x \leq 2\pi$" means only angles in one full turn count.
Intersection method
Reading the solutions as the crossing points of two graphs. Like this: solve $\sin x = 0.5$ by drawing $y = \sin x$ and the line $y = 0.5$ and reading where they meet.
Period
How far along the $x$-axis a graph travels before it repeats. Like this: $y = \sin x$ repeats every $2\pi$, so its solutions repeat every $2\pi$ as well.
Number of solutions
How many crossings fall inside the given domain. Like this: $\sin x = 0.5$ has two solutions in $[0, 2\pi]$, one going up and one coming down.
Related acute angle
The first-quadrant angle a calculator returns, used to find the rest. Like this: $\sin^{-1}(0.5) = \tfrac{\pi}{6}$, and the second solution is $\pi - \tfrac{\pi}{6}$.
4
You’re here
Solving trigonometric equations using graphs
Work through the core explanation before applying it.
05
Solving trigonometric equations using graphs
core concept
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 $x = \frac{4\pi}{3}$.
Why this matters for real-world problems. In physics and engineering, equations like $\sin(\omega t) = 0.8$ describe when a rotating object or oscillating system reaches a certain state. Graphical solutions help engineers quickly estimate these times without solving complicated inverse trig equations, especially when damping or external forces make the equation impossible to solve exactly.
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 $0 \leq x \leq 2\pi$?
06
Counting solutions in an interval
core concept
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
5
You’re here
Work examples end to end
Follow the reasoning through complete worked solutions.
PROBLEM 1 · GRAPHICAL SOLUTION+5 XP on full reveal
By sketching $y = \sin x$ and $y = \cos x$ on the same axes, find all solutions to $\sin x = \cos x$ in $0 \leq x \leq 2\pi$.
1
Sketch both graphs
$y = \sin x$ starts at $(0, 0)$ and peaks at $\frac{\pi}{2}$. $y = \cos x$ starts at $(0, 1)$ and crosses zero at $\frac{\pi}{2}$.
2
$\sin x = \cos x \Rightarrow \tan x = 1$
The graphs intersect where $\sin x = \cos x$, which is where $\tan x = 1$.
3
$x = \frac{\pi}{4}, \frac{5\pi}{4}$
In $[0, 2\pi]$, tangent equals 1 at $\frac{\pi}{4}$ and $\frac{5\pi}{4}$.
PROBLEM 2 · COUNTING SOLUTIONS+5 XP on full reveal
How many solutions does $\sin x = 0.2$ have in $0 \leq x \leq 4\pi$?
1
Determine the number of cycles
In $4\pi$, the sine graph completes 2 full cycles (period $= 2\pi$).
2
Count intersections per cycle
The line $y = 0.2$ intersects each cycle twice (once rising, once falling).
Multiply the number of cycles by the number of intersections per cycle.
PROBLEM 3 · TRANSFORMED EQUATION+5 XP on full reveal
By considering the graphs of $y = 2\sin x$ and $y = 1$, find all solutions to $2\sin x = 1$ in $0 \leq x \leq 2\pi$.
1
$\sin x = \frac{1}{2}$
Rewrite the equation so one side is a single trig function.
2
Sketch $y = \sin x$ and $y = \frac{1}{2}$
The line intersects the sine curve twice in $[0, 2\pi]$.
3
$x = \frac{\pi}{6}, \frac{5\pi}{6}$
The first quadrant solution is $\frac{\pi}{6}$; the second quadrant solution is $\pi - \frac{\pi}{6} = \frac{5\pi}{6}$.
Common errors · the 3 traps that cost marks
6
You’re here
Dodge the traps, then apply
Meet the mistakes that cost marks, then do it yourself.
Trap 01
Only finding one solution when there are two per cycle
For equations like $\sin x = 0.5$, students often find $x = \frac{\pi}{6}$ but forget the second solution in the same cycle ($\frac{5\pi}{6}$). Fix: Sine is positive in Q1 and Q2; cosine is positive in Q1 and Q4. Check all relevant quadrants.
Trap 02
Forgetting to account for multiple cycles
If the domain spans more than one period, there can be more than two solutions. Students sometimes stop after finding solutions in the first period. Fix: Count how many complete cycles fit in the domain, then multiply the number of solutions per cycle accordingly.
Trap 03
Including endpoints that are not in the domain
If the domain is $0 \leq x < 2\pi$, then $x = 2\pi$ is not included. Be careful with strict inequalities. Fix: Always check whether endpoints satisfy the domain restrictions.
Quick-fire practice · 5 reps +2 XP per reveal
7
You’re here
Drill it, then lock it in
Run the quick drill and copy the summary into your book.
1
$\cos x = 0.5$ for $0 \leq x \leq 2\pi$, state all solutions.
2
$\sin x = -0.5$ for $0 \leq x \leq 2\pi$, state all solutions.
3
$\tan x = \sqrt{3}$ for $0 \leq x \leq 2\pi$, state all solutions.
4
How many solutions does $\sin x = 0.7$ have in $[0, 6\pi]$? Explain.
5
By sketching, find all solutions to $\sin x = \cos x$ in $[0, 2\pi]$.
12
Revisit your thinking
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.
auto-saved
1
You’re here
Multiple choice
Answer the drill bank and rate your confidence.
01
Multiple choice
+5 XP per correct · +25 XP all-correct
Pick your answer, then rate your confidence, that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
2
You’re here
Short answer
Write full responses, then check them against the model answers.
02
Short answer
ApplyBand 43 marks
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)
auto-saved
ApplyBand 42 marks
Q2. How many solutions does $\sin x = 0.7$ have in $0 \leq x \leq 6\pi$? Explain your reasoning. (2 marks)
auto-saved
AnalyseBand 53 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)
auto-saved
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].
1
You’re here
Review and finish
Take the module quiz if you are ready, then mark the lesson complete.
01
Boss battle · Graphical Trig Solutions
earn bronze · silver · gold
Five timed questions. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
Mark lesson as complete
Tick when you've finished the practice and review.
