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Using the Addition Formula0%

Trigonometry · Topic 14 of 19

Using the Addition Formula

Video lesson · from 34:462 worked examples

One lesson video covers all of Trigonometry, so it opens at 34:46 for this topic — not from the beginning.

Theory

Given a function of the form asin⁡x+bcos⁡xa\sin x + b\cos x, it is useful to express this as a single function rather than the sum of two separate functions.

A single function would allow us to calculate maximum and minimum values of expressions of the form asin⁡x+bcos⁡xa\sin x + b\cos x, sketch graphs more easily and solve equations involving expressions of this form.

We previously studied the addition formulae, and this will help to express sums of two separate functions as single functions.

⚠️ Common Examiner Traps

  • Expand the target, not the question: write out the addition formula for the form you have been asked for, then compare it with the expression you were given.
  • Match the right terms: line the cos⁡x\cos x terms up with the cos⁡x\cos x terms. Comparing them the wrong way round swaps your aa and bb.
  • Use the form the question specifies: kcos⁡(x−α)k\cos(x-\alpha) and ksin⁡(x+α)k\sin(x+\alpha) give different values of α\alpha. Answering in a different form loses marks even if the maths is sound.
  • Show the comparison step: the marks are for equating the coefficients, so write that line down.

Worked examples

Example 1

Create two equations for 4cos⁡x+3sin⁡x4\cos x + 3\sin x using the expansion for kcos⁡(x−α)k\cos(x - \alpha).

1. Expand kcos⁡(x−α)k\cos(x - \alpha):

kcos⁡(x−α)=kcos⁡xcos⁡α+ksin⁡xsin⁡αk\cos(x - \alpha) = k\cos x\cos\alpha + k\sin x\sin\alpha

2. Equate to the given expression:

kcos⁡αcos⁡x+ksin⁡αsin⁡x=4cos⁡x+3sin⁡xk\cos\alpha\cos x + k\sin\alpha\sin x = 4\cos x + 3\sin x

3. Equate coefficients of cos⁡x\cos x and sin⁡x\sin x:

kcos⁡α=4ksin⁡α=3\begin{aligned} k\cos\alpha &= 4 \\ k\sin\alpha &= 3 \end{aligned}

Example 2

Create two equations for 2sin⁡x−3cos⁡x2\sin x - 3\cos x using the expansion for ksin⁡(x+α)k\sin(x + \alpha).

1. Expand ksin⁡(x+α)k\sin(x + \alpha):

ksin⁡(x+α)=ksin⁡xcos⁡α+kcos⁡xsin⁡αk\sin(x + \alpha) = k\sin x\cos\alpha + k\cos x\sin\alpha

2. Equate to the given expression:

kcos⁡αsin⁡x+ksin⁡αcos⁡x=2sin⁡x−3cos⁡xk\cos\alpha\sin x + k\sin\alpha\cos x = 2\sin x - 3\cos x

3. Equate coefficients of sin⁡x\sin x and cos⁡x\cos x:

kcos⁡α=2ksin⁡α=−3\begin{aligned} k\cos\alpha &= 2 \\ k\sin\alpha &= -3 \end{aligned}