Deriving Compound-Angle Identities

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Deriving Compound-Angle Identities

The sine and cosine of $A\pm B$ come from coordinates or projections, not from distributing the function. Their sign patterns must be preserved carefully.

Worked reasoning

$\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$ .
Substitute exact values to get $(\sqrt6+\sqrt2)/4$ .
This is approximately $0.966$ .

Key idea

$\sin(A+B)=\sin A\cos B+\cos A\sin B$ and $\cos(A+B)=\cos A\cos B-\sin A\sin B$ .

Evaluate $\sin75^\circ$ correct to three decimals using $45^\circ+30^\circ$ .

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

$\sin(A+B)=\sin A\cos B+\cos A\sin B$ and $\cos(A+B)=\cos A\cos B-\sin A\sin B$ .

$\sin(A+B)=\sin A+\sin B$ .

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 Deriving Compound-Angle Identities, which move is most reliable?

Write the full identity before substituting special values.

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 Deriving Compound-Angle Identities?

$\sin(A+B)=\sin A+\sin B$ .

$\sin(A+B)=\sin A\cos B+\cos A\sin B$ and $\cos(A+B)=\cos A\cos B-\sin A\sin B$ .

Write the full identity before substituting special values.

Test the formula at $A=0$ or another exact-angle case.

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