Unit 5 · Topic 3.10

Trigonometric Equations and Inequalities

Combine algebra, inverse functions, periodicity, unit-circle reasoning, and graphs to find every solution allowed by a stated domain.

Learning Goals

Distinguish a trigonometric equation from an identity.
Solve equations exactly or approximately on a specified interval.
Write general solutions using the correct period.
Solve transformed and quadratic-form trigonometric equations.
Represent inequality solutions and apply contextual domain restrictions.

1. Equation, Identity, or Contradiction?

A trigonometric equation is true only for selected inputs, while an identity is true for every input in its domain.

\[\sin x=\frac12\quad\text{is an equation,}\]

\[\sin^2x+\cos^2x=1\quad\text{is an identity.}\]

An equation such as \(\sin x=2\) is a contradiction over the real numbers because sine outputs never exceed 1.

2. Read the Domain before Solving

The requested answer may be all real solutions, solutions on a finite interval, or only values meaningful in a context. Record endpoint inclusion before doing any algebra.

Prompt	Expected answer form
\(0\le x<2\pi\)	A finite list in that interval
All real \(x\)	Families using \(k\in\mathbb Z\)
Time during one cycle	Only contextually meaningful times

3. An Inverse Function Gives a Starting Angle

If \(\sin x=c\), then \(\arcsin c\) gives one principal angle. It does not automatically provide every solution of the periodic equation.

Isolate one trigonometric expression.
Use an inverse function or exact unit-circle value to find a reference angle.
Determine every quadrant with the required sign.
Add periods or filter candidates to the stated interval.
Verify in the original equation.

4. Use Signs to Select Quadrants

Function is positive	Quadrants	Function is negative	Quadrants
Sine	I, II	Sine	III, IV
Cosine	I, IV	Cosine	II, III
Tangent	I, III	Tangent	II, IV

Reference angles give the magnitude; quadrant signs determine the actual angles.

5. Interpret Solutions as Intersections

A horizontal line at \(\frac{1}{2}\) intersects one sine cycle twice.

Solutions of \(f(x)=c\) are the x-coordinates where \(y=f(x)\) intersects the horizontal line \(y=c\). For an inequality, the solution consists of intervals where the trigonometric graph lies above or below that line.

6. Solve an Exact Equation on One Cycle

Solve \(\sin x=1/2\) on \(0\le x<2\pi\). The reference angle is \(\pi/6\), and sine is positive in Quadrants I and II:

\[x=\frac{\pi}{6}\quad\text{or}\quad x=\pi-\frac{\pi}{6}=\frac{5\pi}{6}.\]

Both values lie in the requested interval. Reporting only \(\arcsin(1/2)=\pi/6\) would miss the second intersection.

7. Write General Solutions

Let \(k\in\mathbb Z\). Convenient general forms are

Equation	Principal value	All real solutions
\(\sin x=c\)	\(\alpha=\arcsin c\)	\(x=\alpha+2k\pi\) or \(x=\pi-\alpha+2k\pi\)
\(\cos x=c\)	\(\alpha=\arccos c\)	\(x=\pm\alpha+2k\pi\)
\(\tan x=c\)	\(\alpha=\arctan c\)	\(x=\alpha+k\pi\)

Sine and cosine repeat every \(2\pi\); tangent repeats every \(\pi\).

8. Solve a Transformed Multiple-Angle Equation

Solve \(\sin(2x-\pi/3)=\sqrt3/2\) on \(0\le x<2\pi\). Let \(u=2x-\pi/3\). Then

\[u=\frac{\pi}{3}+2k\pi\quad\text{or}\quad u=\frac{2\pi}{3}+2k\pi.\]

\[x=\frac{\pi}{3}+k\pi\quad\text{or}\quad x=\frac{\pi}{2}+k\pi.\]

Filtering to the given interval gives

\[x=\frac{\pi}{3},\ \frac{\pi}{2},\ \frac{4\pi}{3},\ \frac{3\pi}{2}.\]

The compressed input creates two sine cycles and therefore four solutions on the x-interval.

9. Respect Tangent's Shorter Period

Solve \(\tan(x-\pi/6)=-1\) on \(0\le x<2\pi\):

\[x-\frac{\pi}{6}=-\frac{\pi}{4}+k\pi\quad\Longrightarrow\quad x=-\frac{\pi}{12}+k\pi.\]

The values in the interval are

\[x=\frac{11\pi}{12},\quad\frac{23\pi}{12}.\]

10. Use Algebra on Quadratic Form

Solve \(2\cos^2x-\cos x-1=0\) on \(0\le x<2\pi\):

\[(2\cos x+1)(\cos x-1)=0.\]

\[\cos x=-\frac12\quad\text{or}\quad\cos x=1.\]

Unit-circle values give

\[x=0,\quad\frac{2\pi}{3},\quad\frac{4\pi}{3}.\]

Factoring finds possible trigonometric outputs; each output must still be converted into every angle in the domain.

11. Approximate Non-Special Solutions

Solve \(\cos x=0.2\) on \(0\le x<2\pi\). In radian mode,

\[\alpha=\arccos(0.2)\approx1.3694.\]

Cosine is positive in Quadrants I and IV, so

\[x\approx1.3694\quad\text{or}\quad x\approx2\pi-1.3694=4.9137.\]

Keep extra calculator digits until the final rounding step.

12. Check Feasibility and Extraneous Values

Before searching for angles, compare the isolated output with the function's range. For example, \(3\sin x=4\) has no real solution because it requires \(\sin x=4/3\).

Operations such as squaring, multiplying by an expression that can equal zero, or using reciprocal forms can introduce invalid candidates. Substitute every candidate into the original equation and exclude inputs where the original expression is undefined.

13. Solve Trigonometric Inequalities

Solve \(\sin x>1/2\) on \(0\le x<2\pi\). First solve the boundary equation \(\sin x=1/2\), obtaining \(\pi/6\) and \(5\pi/6\).

The sine graph lies above \(1/2\) between those crossings:

\[x\in\left(\frac{\pi}{6},\frac{5\pi}{6}\right).\]

The endpoints are excluded because the inequality is strict. A non-strict symbol would include boundary values that belong to the domain.

14. Apply a Contextual Restriction

A rider's height during one 40-second observation-wheel revolution is

\[h(t)=14-12\cos\left(\frac{\pi t}{20}\right),\qquad 0\le t\le40.\]

To find when the rider is at least 20 meters high, solve \(h(t)\ge20\), or \(\cos(\pi t/20)\le-1/2\). During this revolution,

\[\frac{40}{3}\le t\le\frac{80}{3}.\]

The periodic equation has infinitely many mathematical solutions, but the ride interval leaves one continuous time window.

15. AP Workflow and Common Errors

Record the domain and angle unit.
Rewrite or factor until one trigonometric expression is isolated.
Check whether the requested output is possible.
Find a reference or principal angle.
Use signs and periodicity to generate all candidates.
Filter endpoints and contextual restrictions.
Verify candidates in the original representation.

Do not stop after the calculator returns one principal angle.
Do not add \(2\pi k\) to tangent solutions.
Do not divide by a trigonometric factor and lose the solutions where that factor is zero.
Do not connect inequality intervals across an undefined input.
Do not ignore open or closed endpoints.

Key Takeaways

Solve trigonometric equations and inequalities using reference angles, identities, graphs, and stated intervals.
Core relationship: \(\sin x=\frac{1}{2}\)
Error check: Do not report only the inverse-trigonometric principal value when the interval contains more solutions.

Checkpoint · Topic 3.10

Solve \(\cos x=-\sqrt2/2\) on \(0\le x<2\pi\).
Write all real solutions of \(\tan x=\sqrt3\).
Solve \(2\sin^2x-3\sin x+1=0\) on \(0\le x<2\pi\).
Solve \(\cos(2x)=0\) on \(0\le x<2\pi\).
Solve \(\cos x\le-1/2\) on \(0\le x\le2\pi\) and explain the endpoint choices.