Equations with Compound and Double Angles

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Equations with Compound and Double Angles

When a trig equation contains $2x$ or a compound expression, solve for that whole angle first over its expanded interval, then divide or isolate $x$ . General-solution periods transform with the angle multiplier.

Worked reasoning

$0^\circ\le2x\le360^\circ$ .
$\sin2x=0$ at $2x=0^\circ,180^\circ,360^\circ$ , so $x=0^\circ,90^\circ,180^\circ$ .

Key idea

Solve for the entire angle expression before converting back to $x$ .

For $0^\circ\le x\le180^\circ$ , which solutions satisfy $\sin2x=0$ ?

$x=0^\circ,90^\circ,180^\circ$

$x=0^\circ,180^\circ$

$x=90^\circ$ only

$x=45^\circ,135^\circ

Which statement best captures the central mathematical idea in Equations with Compound and Double Angles?

Solve for the entire angle expression before converting back to $x$ .

The solution range stays unchanged when replacing $x$ by $2x$ .

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 Equations with Compound and Double Angles, which move is most reliable?

Determine the range of the compound angle from the stated range of $x$ .

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 Equations with Compound and Double Angles?

The solution range stays unchanged when replacing $x$ by $2x$ .

Solve for the entire angle expression before converting back to $x$ .

Determine the range of the compound angle from the stated range of $x$ .

Substitute each final $x$ into the original equation, not only the reduced one.

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