Sum and difference formulas
Sum and Difference Formulas
Sum and Difference Formulas is a topic in Trigonometric Identities in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.5(M), which requires students to simplify trigonometric expressions using the sum and difference identities.
The sum and difference formulas expand \(\sin\), \(\cos\), and \(\tan\) of \(A\pm B\), giving exact values of non-special angles such as \(75^\circ\).
Theory
The sum and difference formulas expand a trig function of a combined angle \(A\pm B\) in terms of functions of \(A\) and \(B\) separately. They let you find exact values of non-special angles (like \(75^\circ=45^\circ+30^\circ\)) and simplify expressions such as \(\cos(\pi+x)\).
The six formulas (top sign with \(+\), bottom with \(-\)):
How to use a sum or difference formula
- Split the angle into a sum or difference of special angles.
- Pick the formula for the function you need.
- Substitute the exact values and mind the signs.
- Simplify, rationalizing if a radical lands in a denominator.
Write \(75^\circ=45^\circ+30^\circ\) and use the cosine sum formula.
| \(\cos 75^\circ\) | \(=\) | \(\cos 45^\circ\cos 30^\circ-\sin 45^\circ\sin 30^\circ\) |
| \(=\) | \(\dfrac{\sqrt2}{2}\cdot\dfrac{\sqrt3}{2}-\dfrac{\sqrt2}{2}\cdot\dfrac{1}{2}\) | |
| \(=\) | \(\dfrac{\sqrt6-\sqrt2}{4}\) |
Apply the sine difference formula.
| \(\sin(x-\dfrac{\pi}{2})\) | \(=\) | \(\sin x\cos\dfrac{\pi}{2}-\cos x\sin\dfrac{\pi}{2}\) |
| \(=\) | \(\sin x\cdot 0-\cos x\cdot 1\) | |
| \(=\) | \(-\cos x\) |
Use the tangent sum formula with \(\tan 45^\circ=1\), \(\tan 30^\circ=\dfrac{1}{\sqrt3}\).
| \(\tan 75^\circ\) | \(=\) | \(\dfrac{1+\dfrac{1}{\sqrt3}}{1-1\cdot\dfrac{1}{\sqrt3}}\) |
| \(=\) | \(\dfrac{\sqrt3+1}{\sqrt3-1}=2+\sqrt3\) |
Use the cosine sum formula with \(A=\pi\).
| \(\cos(\pi+x)\) | \(=\) | \(\cos\pi\cos x-\sin\pi\sin x\) |
| \(=\) | \((-1)\cos x-0\) | |
| \(=\) | \(-\cos x\) |
Common pitfalls
Frequently asked questions
What is the cosine sum formula?
\(\cos(A+B)=\cos A\cos B-\sin A\sin B\). Note the minus sign, opposite the outer plus.
Why can't you just write sin(A+B) = sinA + sinB?
Because sine is not linear. The correct expansion mixes both angles: \(\sin A\cos B+\cos A\sin B\).
How do you find cos 75 degrees exactly?
Write \(75^\circ=45^\circ+30^\circ\) and apply the cosine sum formula to get \(\dfrac{\sqrt6-\sqrt2}{4}\).
How do you remember the signs?
Sine keeps the sign (plus stays plus); cosine flips it (plus becomes minus). Tangent flips the sign in its denominator.