Advanced trigonometric equations

Objective. Solve transformed equations over exact intervals.

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Learn: the key idea

Advanced trigonometric equations combine identities such as double-angle or harmonic form with a restricted interval, requiring all valid solutions to be found exactly within that range. Example: solving cos2x=0.5 for 0≤x≤360° gives 2x=60°,300°,420°,660°, so x=30°,150°,210°,330° after dividing by 2 and adjusting for the doubled range. Always extend the interval appropriately before dividing back down when the equation involves a multiple angle.

Worked example

Solve the trigonometric equation sin2x=√3/2 for 0≤x≤360°.

1 $Extend the interval: 0\leq2x\leq720°$
2 $Find base solutions: 2x=60°,120°$
3 $Add 360° to each for the extended range: 2x=420°,480°$
4 $Divide all four values by 2: x=30°,60°,210°,240°$

Answer: x=30°, 60°, 210°, 240°

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Common mistake. Students solve 2x=60° for x but forget to extend the interval to 0≤2x≤720° first, missing extra solutions that arise from the doubled angle range.

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