LEARN · EXPLAIN · REVISE

A student claims sin(x+π)=sin x for all real x. Give a counterexample and state the correct identity.

Read the idea, work independently, then explain what changed.

TOPIC 01

Trigonometric ratios and function graphs

A half-turn changes the sign; the full period of sine is 2π.

PREDICT → MOVE → EXPLAIN

Predict the number of cycles and the horizontal shift. Switch angle units without changing the curve.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

\sin(x+\pi)=\sin x ?

Working and explanation

BUILD THE REASONING

Hint 1

Test an angle whose sine is nonzero.

Hint 2

Try x=π/2, then use the angle-addition formula.

Worked solution

One valid counterexample disproves an all-real claim.
$\sin(3\pi/2)=-1\ne1=\sin(\pi/2)$
Apply the compound-angle identity.
$\sin(x+\pi)=\sin x\cos\pi+\cos x\sin\pi=-\sin x$

Counterexample x=π/2; correct identity sin(x+π)=−sin x.

Checks and common pitfalls: A half-turn changes the sign; the full period of sine is 2π.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.