Basic trigonometric equations

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The problem that opens the lesson

Solve sin $x=\frac{1}{2}$ for all real $x$ and then list solutions on $[-pi,3pi]$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. State general solutions with an integer parameter, then filter them into a requested interval. Keep degree or radian mode consistent. The relevant conditions are not optional bookkeeping: Sine and cosine have no real solution when $|k|>1$ . Reciprocal functions impose additional domain exclusions. Following that structure gives $x=\frac{pi}{6}+2kpi$ or $\frac{5pi}{6}+2kpi$ ; interval solutions follow from those families.

Why this works

Exact unit-circle values should be used when possible. Otherwise find a principal inverse angle and use graph symmetry to generate the remaining branch before adding periods. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

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What this lesson is really about

A basic trigonometric equation asks for every input whose function value equals a given constant.

On one period, a horizontal line may meet sine or cosine twice and tangent once. Periodicity then generates all real solutions.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

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Why the relationship works

Exact unit-circle values should be used when possible. Otherwise find a principal inverse angle and use graph symmetry to generate the remaining branch before adding periods.

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A reliable way to work

State general solutions with an integer parameter, then filter them into a requested interval. Keep degree or radian mode consistent.

Sine and cosine have no real solution when $|k|>1$ . Reciprocal functions impose additional domain exclusions.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

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What commonly goes wrong

A common error is to report only the calculator’s principal inverse value.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

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Worked examples

Worked example 1

Solve sin $x=\frac{1}{2}$ for all real $x$ and then list solutions on $[-pi,3pi]$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. State general solutions with an integer parameter, then filter them into a requested interval. Keep degree or radian mode consistent. The relevant conditions are not optional bookkeeping: Sine and cosine have no real solution when $|k|>1$ . Reciprocal functions impose additional domain exclusions. Following that structure gives $x=\frac{pi}{6}+2kpi$ or $\frac{5pi}{6}+2kpi$ ; interval solutions follow from those families.

Why this works

Exact unit-circle values should be used when possible. Otherwise find a principal inverse angle and use graph symmetry to generate the remaining branch before adding periods. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Solve cos $x=-\frac{sqrt(2)}{2}$

Worked development

State general solutions with an integer parameter, then filter them into a requested interval. Keep degree or radian mode consistent. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. On one period, a horizontal line may meet sine or cosine twice and tangent once. Periodicity then generates all real solutions. Then apply the conditions explicitly: Sine and cosine have no real solution when $|k|>1$ . Reciprocal functions impose additional domain exclusions. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Basic equations locate times, angles, phases, and intersections in periodic models.

Reasoning example

Problem

Solve tan $x=3$ numerically.

Worked development

State general solutions with an integer parameter, then filter them into a requested interval. Keep degree or radian mode consistent. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. On one period, a horizontal line may meet sine or cosine twice and tangent once. Periodicity then generates all real solutions. Then apply the conditions explicitly: Sine and cosine have no real solution when $|k|>1$ . Reciprocal functions impose additional domain exclusions. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Basic equations locate times, angles, phases, and intersections in periodic models.

Worked example 4: quick check

Solve cos $x=0$ on $[0,2pi]$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. State general solutions with an integer parameter, then filter them into a requested interval. Keep degree or radian mode consistent. The relevant conditions are not optional bookkeeping: Sine and cosine have no real solution when $|k|>1$ . Reciprocal functions impose additional domain exclusions. Following that structure gives $x=\frac{pi}{2}$ and $\frac{3pi}{2}$ .

Why this works

Exact unit-circle values should be used when possible. Otherwise find a principal inverse angle and use graph symmetry to generate the remaining branch before adding periods. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

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Application and interpretation

Basic equations locate times, angles, phases, and intersections in periodic models.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice

Ten concrete questions

Practice 101

Solve cos $x=0$ on $[0,2pi]$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer