Double-Angle Identities

≈ 25 min

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Double-Angle Identities

Set $B=A$ in compound-angle formulas to obtain $\sin2A=2\sin A\cos A$ and three equivalent forms of $\cos2A$ . Choose the cosine form that matches the information provided.

Worked reasoning

Use $\cos2A=1-2\sin^2A$ .
$1-2(9/25)=1-18/25=7/25$ .
$7/25=0.28$ .

Key idea

Equivalent double-angle forms are selected to eliminate the unknown ratio.

If $\sin A=3/5$ and $A$ is acute, find $\cos2A$ .

Which statement best captures the central mathematical idea in Double-Angle Identities?

Equivalent double-angle forms are selected to eliminate the unknown ratio.

$\cos2A$ has only one valid form, $2\sin A\cos A$ .

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Double-Angle Identities, which move is most reliable?

List the known trigonometric ratio and quadrant before choosing an identity.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Double-Angle Identities?

$\cos2A$ has only one valid form, $2\sin A\cos A$ .

Equivalent double-angle forms are selected to eliminate the unknown ratio.

List the known trigonometric ratio and quadrant before choosing an identity.

Check sign and range against the quadrant of $2A$ .

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