Change the form, keep the function

Compound-angle identities connect shifted waves with sine and cosine. They let you combine a $\sin x$ + b $\cos x$ into one wave with an amplitude and phase shift.

Reciprocal trigonometric functions are sec x = 1/cos x, cosec x = 1/sin x and cot x = $\cos x$ /sin x where the denominators are nonzero. Pythagorean and compound-angle identities let you change form without changing a function on its domain.

\displaystyle \sin(a+b)=\sin a\cos b+\cos a\sin b

WORKED EXAMPLE

Express 3 $\sin x$ + 4 $\cos x$ as R sin(x + α), with R > 0 and α acute.

Expand R sin(x + α) = R cos α $\sin x$ + R sin α $\cos x$ .
Match coefficients: R cos α = 3 and R sin α = 4.
Thus R = 5 and α = arctan( $\frac{4}{3}$ ).

Assumed knowledge

Use radians, the unit circle, exact sine and cosine values, and their graphs. Recall the basic Pythagorean identity, algebraic expansion, and restrictions caused by zero denominators.

Study Specialist Mathematics alongside, or after, Mathematical Methods. Later units build on earlier units.

QCAA prior-learning guidance ↗

Change the form, keep the function

Express 3 sin⁡x\sin xsinx + 4 cos⁡x\cos xcosx as R sin(x + α), with R > 0 and α acute.

Put the idea to work.

Express 3 $\sin x$ + 4 $\cos x$ as R sin(x + α), with R > 0 and α acute.