Double-angle formulas

Textbook reading

The problem that opens the lesson

Given sin $theta=\frac{3}{5}$ in quadrant II, find sin $2theta$ and cos $2theta$ exactly.

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine the quadrant of $x$ and $2x$ when signs matter, select the form matching known data, and preserve exact values. The relevant conditions are not optional bookkeeping: Knowing sin $x$ alone may not determine cos $x$ without quadrant information. Following that structure gives sin $2theta=-\frac{24}{25}$ ; cos $2theta=\frac{7}{25}$ .

Why this works

Different cosine forms are useful depending on which function must be eliminated. 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

Double-angle formulas describe trig values at $2x$ using values at $x$ .

Substituting $alpha=beta=x$ into the sum formulas gives $sin2x=2sin x$ cos $x$ and $cos2x=cos^2 x-sin^2 x$ . Pythagorean substitution creates the alternative cosine forms $1-2sin^2 x$ and $2cos^2 x-1$ .

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

Different cosine forms are useful depending on which function must be eliminated.

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A reliable way to work

Determine the quadrant of $x$ and $2x$ when signs matter, select the form matching known data, and preserve exact values.

Knowing sin $x$ alone may not determine cos $x$ without quadrant information.

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 interpret $sin2x$ as $2sin x$ or to square the angle instead of the function 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

Given sin $theta=\frac{3}{5}$ in quadrant II, find sin $2theta$ and cos $2theta$ exactly.

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine the quadrant of $x$ and $2x$ when signs matter, select the form matching known data, and preserve exact values. The relevant conditions are not optional bookkeeping: Knowing sin $x$ alone may not determine cos $x$ without quadrant information. Following that structure gives sin $2theta=-\frac{24}{25}$ ; cos $2theta=\frac{7}{25}$ .

Why this works

Different cosine forms are useful depending on which function must be eliminated. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Derive sin $2x$ from the sum formula.

Worked development

Determine the quadrant of $x$ and $2x$ when signs matter, select the form matching known data, and preserve exact values. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Substituting $alpha=beta=x$ into the sum formulas gives $sin2x=2sin x$ cos $x$ and $cos2x=cos^2 x-sin^2 x$ . Pythagorean substitution creates the alternative cosine forms $1-2sin^2 x$ and $2cos^2 x-1$ . Then apply the conditions explicitly: Knowing sin $x$ alone may not determine cos $x$ without quadrant information. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Double angles appear in geometry, signal harmonics, power reduction, and equation solving.

Reasoning example

Problem

Compare $1-2sin^2 x \qquad 2cos^2 x-1$

Worked development

Determine the quadrant of $x$ and $2x$ when signs matter, select the form matching known data, and preserve exact values. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. Substituting $alpha=beta=x$ into the sum formulas gives $sin2x=2sin x$ cos $x$ and $cos2x=cos^2 x-sin^2 x$ . Pythagorean substitution creates the alternative cosine forms $1-2sin^2 x$ and $2cos^2 x-1$ . Then apply the conditions explicitly: Knowing sin $x$ alone may not determine cos $x$ without quadrant information. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Double angles appear in geometry, signal harmonics, power reduction, and equation solving.

Worked example 4: quick check

Rewrite cos $2x$ entirely in terms of sin $x$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. Determine the quadrant of $x$ and $2x$ when signs matter, select the form matching known data, and preserve exact values. The relevant conditions are not optional bookkeeping: Knowing sin $x$ alone may not determine cos $x$ without quadrant information. Following that structure gives $1-2sin^2 x$ .

Why this works

Different cosine forms are useful depending on which function must be eliminated. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Double angles appear in geometry, signal harmonics, power reduction, and equation solving.

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