Harmonic form

Objective. Write a cos x+b sin x as R cos(x−α).

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Harmonic form rewrites a combination acos x+bsin x as a single wave Rcos(x−α), where R=√(a²+b²) and α=arctan(b/a), simplifying maximum/minimum finding and equation solving. Example: 3cos x+4sin x=5cos(x−53.1°), since R=√(3²+4²)=5 and tanα=4/3 gives α=53.1°. The maximum value of the expression is then simply R, occurring when x−α=0.

Worked example

Write 5cos x+12sin x in harmonic form Rcos(x−α), giving R exactly and α to 1 decimal place.

1 $Calculate R=\sqrt(5²+12²)=\sqrt169=13$
2 $Calculate α using tanα=12/5$
3 $Evaluate α=arctan(2.4)=67.4°$
4 $State the harmonic form using R and α$

Answer: 13cos(x−67.4°)

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students forget to check the correct quadrant for α when b is negative, giving an angle with the wrong sign; always sketch or consider signs of a and b to place α correctly.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard