Solve a circular equation on a finite domain
First isolate the circular-function value, then generate all symmetric and periodic solutions. Keep only the values in the stated interval and count a shared endpoint once
\[T_{\sin}=T_{\cos}=2\pi,\quad T_{\tan}=\pi\]
- Isolate the function Rearrange the equation until the sine, cosine or tangent value is alone \(\displaystyle 2\sin x+1=0\Rightarrow\sin x=-\frac{1}{2}\)
- Check the function range A sine or cosine value must lie between −1 and 1 before solutions are kept \(\displaystyle -1\le\sin x\le1\)
- Include symmetric solutions Use the relevant quadrant pattern and period to list every angle with the target value \(\displaystyle \sin x=\frac{1}{2},\quad x=\frac{\pi}{6},\frac{5\pi}{6}\text{ on }[0,2\pi)\)
- Filter the stated interval Retain only solutions in the given domain and do not list a shared endpoint twice \(\displaystyle x\in[0,2\pi)\Rightarrow\text{retain each solution once}\)
Worked example
Solve the equation on the stated domain
\[3\cos x-\frac{3}{2}=0,\quad0\le x<2\pi\]
- \(\displaystyle 3\cos x=\frac{3}{2}\)Add the constant term to both sides
- \(\displaystyle \cos x=\frac{1}{2}\)Divide both sides by 3
- \(\displaystyle \alpha=\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\)The reference angle follows from the displayed cosine value
- \(\displaystyle x=\frac{\pi}{3},\quad2\pi-\frac{\pi}{3}=\frac{5\pi}{3}\)Cosine is positive in quadrants I and IV, and both angles lie in the domain
Common mistakes
Applying the inverse too early
✕\(\displaystyle 2\sin x+1=0\Rightarrow x=\sin^{-1}(0)\)✓\(\displaystyle 2\sin x+1=0\Rightarrow\sin x=-\frac{1}{2}\)
The circular-function value must be isolated before applying an inverse functionOmitting a symmetric solution
✕\(\displaystyle \sin x=\frac{1}{2},\quad x=\frac{\pi}{6}\)✓\(\displaystyle \sin x=\frac{1}{2},\quad x=\frac{\pi}{6},\frac{5\pi}{6}\text{ on }[0,2\pi)\)
Sine has the same positive value in quadrants I and II, so both angles are needed in one periodCounting a boundary twice
✕\(\displaystyle \sin x=0\text{ on }[0,2\pi]\Rightarrow x=0,0,\pi,2\pi\)✓\(\displaystyle \sin x=0\text{ on }[0,2\pi]\Rightarrow x=0,\pi,2\pi\)
The shared endpoint represents one angle and must be counted onceDifficulty levels
introSolve an isolated sine, cosine or tangent equation on a standard finite interval
standardRearrange an equation and apply boundary or approximate-angle checks in the domain
challengeSolve an equation involving a square, quadratic, ratio or affine argument
examSolve the full equation, including all valid branches, periods and domain restrictions
Questions about this topic
Why can a circular equation have more than one solution?+
Sine, cosine and tangent repeat and can take the same value at different angles, so every matching angle in the stated domain must be listed
How does the interval affect the answer?+
The interval decides which periodic solutions are retained. An endpoint included by the interval is counted once, even if it is also reached by the next cycle
Why must the circular function be isolated first?+
Inverse functions and quadrant patterns apply to the circular-function value. Rearranging first prevents the coefficient or constant from being treated as part of the angle
Is my complete written rule automatically marked?+
No. Written rules and explanations are ungraded
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 1: Solving equations
In this chapter
Factorise, solve and reconstruct a rational quadraticSolve a quadratic exactly with the quadratic formulaSolve an equation that reduces to quadratic formDetermine parameter values from a discriminant conditionDivide polynomials and state quotient and remainderApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsDetermine parameters in a polynomial identityUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityUse exact trigonometric identities and quadrant informationSolve circular equations over finite domainsGeneral solutions of circular equations and interval extractionLocate and bisect a circular-equation rootRearrange literal equations and recognise equivalent line formsSolve a pair of simultaneous linear equationsSolve a linear system with three or four unknownsVerify coordinate incidence and determine a parameterSolve literal equations and simultaneous literal equationsClassify and parameterise linear systems