Sum and Difference Formulas

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Sum and Difference Formulas: meaning, method and solved example (Precalculus) – Maths PDF

In short: The sum and difference formulas write sin, cos and tan of A + B or A – B in terms of the trig values of A and B, which unlocks exact values for angles like 75° and 105°.

PrecalculusTrigonometryF-TF.C.9

What is Sum and Difference Formulas?

The sum and difference formulas let you split an unfamiliar angle into two familiar ones. Since 105° is not on the standard list but 60° and 45° are, you can build its sine from theirs.

The identities are \(\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B\); \(\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B\); and \(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}\). Note that the cosine formula flips the sign, and that \(\sin(A + B)\) is never \(\sin A + \sin B\).

How to spot it

  • An exact value is needed for an angle such as 15°, 75° or 105° that is not in the usual table, so sum and difference formulas apply.
  • An angle can be written as a sum or difference of 30°, 45°, 60° or 90° multiples.
  • An expression looks like sin x cos 20° + cos x sin 20° and might collapse to one sine.
  • A proof or simplification involves a shifted angle such as sin(x + π/2).

The method

  1. Break the target angle into two angles with known exact values, such as \(105^{\circ} = 60^{\circ} + 45^{\circ}\).
  2. Write the correct one of the sum and difference formulas and watch the sign pattern, especially for cosine.
  3. Substitute the exact sine and cosine values of both angles.
  4. Multiply, combine over a common denominator, and simplify the radicals.
45°60°105°sin 105° = sin(60° + 45°)= sin 60° cos 45° + cos 60° sin 45°= (√3/2)(√2/2) + (1/2)(√2/2)= (√6 + √2)/4≈ 0.9659
105° = 60° + 45°, so sin 105° = (√6 + √2)/4.

Sum and Difference Formulas: solved example

Problem

Find the exact value of \(\sin 105^{\circ}\) and of \(\cos 75^{\circ}\).

Solution

Write \(105^{\circ} = 60^{\circ} + 45^{\circ}\): \(\sin 105^{\circ} = \frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2} + \frac{1}{2}\cdot\frac{\sqrt{2}}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\).

Write \(75^{\circ} = 45^{\circ} + 30^{\circ}\): \(\cos 75^{\circ} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} – \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6} – \sqrt{2}}{4}\). Results: \(\frac{\sqrt{6} + \sqrt{2}}{4}\) and \(\frac{\sqrt{6} – \sqrt{2}}{4}\).

Try it

Question 1

Find the exact value of \(\cos 15^{\circ}\).

Show the answer

\(\cos(45^{\circ} – 30^{\circ}) = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4}\).

Question 2

Find \(\tan 75^{\circ}\) exactly.

Show the answer

\(\frac{1 + \frac{1}{\sqrt{3}}}{1 – \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} – 1} = 2 + \sqrt{3}\).

Sum and Difference Formulas vs. Double-Angle Formulas

These rules combine two possibly different angles A and B. The doubling rules are what you get when B equals A, so they are special cases. If a problem gives 2θ, use the doubling form; if it gives something like 50° + 20° or x – 30°, use these.

Double-Angle Formulas

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