Table of Contents
In this chapter, we focus on specific trigonometric identities that express the sine, cosine, and tangent of sums and differences of angles, such as $A+B$ or $A-B$. These are called the sum and difference formulas.
They allow you to rewrite expressions like $\sin(75^\circ)$ or $\cos(\alpha+\beta)$ in terms of sines and cosines of simpler angles, and they are essential tools for later topics such as exact values, identities, and trigonometric equations.
The core sum and difference formulas
We will use $A$ and $B$ as general angle variables.
Cosine of a sum and a difference
The cosine formulas are:
$$
\cos(A + B) = \cos A \cos B - \sin A \sin B
$$
$$
\cos(A - B) = \cos A \cos B + \sin A \sin B
$$
Notice the pattern:
- The two formulas look almost the same,
- Only the sign between the two terms changes,
- For cosine, the sign in the formula is the opposite of the sign between $A$ and $B$:
- $+$ in the angle $\Rightarrow$ $-$ in the formula,
- $-$ in the angle $\Rightarrow$ $+$ in the formula.
This “cosine is the opposite sign” rule is a common memory aid.
Sine of a sum and a difference
The sine formulas are:
$$
\sin(A + B) = \sin A \cos B + \cos A \sin B
$$
$$
\sin(A - B) = \sin A \cos B - \cos A \sin B
$$
Key patterns to notice:
- The terms are products of one sine and one cosine.
- For sine, the sign in the formula matches the sign between $A$ and $B$:
- $+$ in the angle $\Rightarrow$ $+$ in the formula,
- $-$ in the angle $\Rightarrow$ $-$ in the formula.
- The two terms are “swapped” versions of each other: $\sin A\cos B$ and $\cos A\sin B$.
Tangent of a sum and a difference
The tangent formulas are:
$$
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
$$
$$
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
$$
Here, the pattern is:
- The sign in the numerator matches the sign between $A$ and $B$.
- The sign in the denominator is the opposite:
- $+$ in the angle $\Rightarrow$ $+$ in numerator, $-$ in denominator,
- $-$ in the angle $\Rightarrow$ $-$ in numerator, $+$ in denominator.
These formulas require that the denominators are not zero (for example, $\tan(A+B)$ is undefined if $1 - \tan A \tan B = 0$).
Using sum and difference formulas for exact values
A common use of these formulas is to find exact trigonometric values for angles that can be written as sums or differences of “special angles” (like $30^\circ$, $45^\circ$, $60^\circ$) whose sine and cosine you already know.
Example: $\sin(75^\circ)$
Write $75^\circ$ as a sum of familiar angles. One choice is:
$$
75^\circ = 45^\circ + 30^\circ
$$
Use the sine sum formula:
$$
\sin(75^\circ) = \sin(45^\circ + 30^\circ)
= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ
$$
Substitute known exact values:
- $\sin 45^\circ = \dfrac{\sqrt{2}}{2}$,
- $\cos 45^\circ = \dfrac{\sqrt{2}}{2}$,
- $\sin 30^\circ = \dfrac{1}{2}$,
- $\cos 30^\circ = \dfrac{\sqrt{3}}{2}$.
So
$$
\sin(75^\circ)
= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}
+ \frac{\sqrt{2}}{2}\cdot\frac{1}{2}
= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
= \frac{\sqrt{6} + \sqrt{2}}{4}.
$$
Example: $\cos(15^\circ)$
Write $15^\circ$ as a difference of familiar angles:
$$
15^\circ = 45^\circ - 30^\circ
$$
Use the cosine difference formula:
$$
\cos(15^\circ) = \cos(45^\circ - 30^\circ)
= \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ
$$
Substitute known values:
$$
\cos(15^\circ)
= \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}
+ \frac{\sqrt{2}}{2}\cdot\frac{1}{2}
= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
= \frac{\sqrt{6} + \sqrt{2}}{4}.
$$
Example: $\tan(75^\circ)$
Again, $75^\circ = 45^\circ + 30^\circ$. Use the tangent sum formula:
$$
\tan(75^\circ) = \tan(45^\circ + 30^\circ)
= \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}.
$$
Use known values:
- $\tan 45^\circ = 1$,
- $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$ (or $\dfrac{\sqrt{3}}{3}$).
Then
$$
\tan(75^\circ)
= \frac{1 + \dfrac{1}{\sqrt{3}}}{1 - 1\cdot\dfrac{1}{\sqrt{3}}}
= \frac{\dfrac{\sqrt{3} + 1}{\sqrt{3}}}{\dfrac{\sqrt{3} - 1}{\sqrt{3}}}
= \frac{\sqrt{3} + 1}{\sqrt{3} - 1}.
$$
You can simplify further by rationalizing the denominator if desired.
Deriving related identities from sum and difference formulas
Once you know these formulas, you can obtain other useful identities by choosing particular values of $A$ or $B$.
Addition of a special angle like $90^\circ$ or $\dfrac{\pi}{2}$
For example, using radians and $B = \dfrac{\pi}{2}$:
$$
\sin\!\left(\frac{\pi}{2} - A\right)
= \sin\frac{\pi}{2}\cos A - \cos\frac{\pi}{2}\sin A
= 1\cdot\cos A - 0\cdot\sin A
= \cos A.
$$
This kind of reasoning connects the sum and difference formulas with co-function identities (which are treated elsewhere), but the step that makes it work is exactly the substitution into the sum/difference formulas.
Double-angle formulas as a special case
If you set $B = A$, the sum formulas give you double-angle identities. For example:
$$
\cos(2A) = \cos(A + A) = \cos^2 A - \sin^2 A,
$$
and
$$
\sin(2A) = \sin(A + A) = 2\sin A\cos A.
$$
The full study of double-angle and related formulas belongs in its own place; here, the important idea is that they come from the sum and difference formulas by choosing $A = B$.
Strategy for choosing $A$ and $B$
When you apply these formulas, most of the work is actually in choosing $A$ and $B$ wisely:
- Try to write the given angle as a sum or difference of angles with known values:
- For degrees: typical building blocks are $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$.
- For radians: typical building blocks are $0$, $\dfrac{\pi}{6}$, $\dfrac{\pi}{4}$, $\dfrac{\pi}{3}$, $\dfrac{\pi}{2}$.
- You can sometimes use more than one option; choose the one that looks simplest.
Examples of decompositions:
- $105^\circ = 60^\circ + 45^\circ$,
- $15^\circ = 45^\circ - 30^\circ$,
- $7\pi/12 = \pi/3 + \pi/4$,
- $-\dfrac{\pi}{12} = \dfrac{\pi}{6} - \dfrac{\pi}{4}$.
Once you have $A$ and $B$, apply the relevant formula, then substitute known exact values.
Common algebraic pitfalls
Using sum and difference formulas is partly trigonometry, partly algebra. Typical mistakes to watch for include:
- Mixing up the signs:
- For cosine, the sign flips.
- For sine and tangent, the sign matches in the numerator (for tangent, denominator flips).
- Forgetting parentheses in tangent expressions:
- Write $\dfrac{\tan A + \tan B}{1 - \tan A\tan B}$, not $\tan A + \dfrac{\tan B}{1 - \tan A\tan B}$.
- Dropping a factor:
- In the sine formulas, every term is a product of a sine and a cosine.
Careful substitution and simplification step by step helps avoid these issues.
Summary
In this chapter, you learned the key sum and difference formulas:
- For cosine:
$$
\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
$$ - For sine:
$$
\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
$$ - For tangent:
$$
\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}
$$
You also saw how to:
- Use these formulas to find exact values for non-standard angles.
- Recognize how other identities (like double-angle and certain co-function relationships) arise as special cases.
These formulas are central tools in trigonometry and will be used repeatedly in simplifying expressions, proving identities, and solving trigonometric equations.