Sum and Difference Formulas

You might think that if you know how to find $\sin 45°$ and $\sin 30°$, then finding $\sin 75°$ should be easy. After all, $75° = 45° + 30°$, so maybe $\sin 75° = \sin 45° + \sin 30°$? It seems reasonable. Addition is addition, right?

Try it. You know that $\sin 45° = \frac{\sqrt{2}}{2} \approx 0.707$ and $\sin 30° = \frac{1}{2} = 0.5$. Adding these gives about $1.207$. But wait. The sine of any angle must be between $-1$ and $1$. So something has gone terribly wrong.

Here is the uncomfortable truth: $\sin(A + B) \neq \sin A + \sin B$. Trigonometric functions do not distribute over addition the way you might hope. If this seems unfair, know that you are not alone. Every student who has ever taken trigonometry has felt the same frustration. But here is the good news: there are formulas that tell you exactly how to handle sums and differences of angles. They are not obvious, but once you learn them, you gain the power to find exact values for angles like $15°$ and $75°$ that do not appear on the unit circle.

And here is the thing: if you already know the basic trig values for $30°$, $45°$, and $60°$, you are most of the way there. The sum and difference formulas just show you how to combine what you already know.

Core Concepts

Why the Simple Approach Fails

Before diving into the formulas, let us understand why $\sin(A + B) \neq \sin A + \sin B$.

Think about what sine actually measures: the $y$-coordinate of a point on the unit circle after rotating by a certain angle. When you rotate by angle $A$ and then by angle $B$, you are not simply adding $y$-coordinates. You are performing a geometric transformation where the second rotation builds on the first in a more complex way.

Consider a concrete example. We know:

• $\sin 30° = 0.5$
• $\sin 60° = \frac{\sqrt{3}}{2} \approx 0.866$

If sine distributed over addition, then $\sin 90°$ would equal $\sin 30° + \sin 60° \approx 0.5 + 0.866 = 1.366$. But $\sin 90° = 1$, not $1.366$. The simple approach simply does not match reality.

This is why we need the sum and difference formulas: they capture the actual geometry of how angles combine.

The Sum Formulas

Here are the formulas for the sum of two angles:

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$

$$\cos(A + B) = \cos A \cos B - \sin A \sin B$$

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

Take a moment to look at these. Notice that each formula involves both angles $A$ and $B$, and it mixes sine and cosine together. This mixing is the key: when you combine rotations, what matters is not just the $y$-coordinates (sines) or just the $x$-coordinates (cosines), but how they interact.

A helpful way to remember the sine formula: it has a plus sign in the middle, matching the plus sign in $\sin(A + B)$. The formula has a symmetric structure: sine-cosine plus cosine-sine.

For cosine, notice the minus sign in the middle. This might seem counterintuitive for a “sum” formula, but it arises from the geometry of rotation.

The tangent formula can actually be derived from the sine and cosine formulas (since $\tan = \frac{\sin}{\cos}$), but it is useful enough to memorize on its own.

The Difference Formulas

The difference formulas are almost identical to the sum formulas, but with the signs flipped in the middle:

$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

$$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Notice the pattern:

• For sine, the sign in the formula matches the operation: plus for sum, minus for difference
• For cosine, the sign in the formula is opposite: minus for sum, plus for difference
• For tangent, the numerator sign matches (plus or minus), while the denominator sign is opposite

If you can remember the sum formulas, you can quickly derive the difference formulas by just flipping the appropriate signs.

Finding Exact Values for Non-Standard Angles

One of the most powerful applications of these formulas is finding exact values for angles that are not on the standard unit circle. Angles like $15°$, $75°$, $105°$, and so on can be expressed as sums or differences of angles you already know.

For example:

• $75° = 45° + 30°$
• $15° = 45° - 30°$ (or $60° - 45°$)
• $105° = 60° + 45°$

This means that if you know the exact trig values for $30°$, $45°$, and $60°$, you can find exact values for many more angles.

Let us recall those values:

Angle	$\sin$	$\cos$	$\tan$
$30°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45°$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

With these values and the sum and difference formulas, a whole new set of angles becomes accessible.

Working with Radians

Everything we have said applies equally to radians. Common combinations include:

• $\frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6}$ (equivalent to $15°$)
• $\frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}$ (equivalent to $75°$)
• $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$ (equivalent to $105°$)

The formulas work the same way regardless of whether you measure angles in degrees or radians.

Simplifying Expressions

The sum and difference formulas can also simplify expressions involving trig functions. For instance, if you encounter $\sin(x + \frac{\pi}{2})$, you can expand it:

$$\sin\left(x + \frac{\pi}{2}\right) = \sin x \cos\frac{\pi}{2} + \cos x \sin\frac{\pi}{2}$$

Since $\cos\frac{\pi}{2} = 0$ and $\sin\frac{\pi}{2} = 1$:

$$= \sin x \cdot 0 + \cos x \cdot 1 = \cos x$$

This shows that $\sin(x + \frac{\pi}{2}) = \cos x$, a cofunction identity that you might already know. The sum formulas let you derive such identities rather than memorizing them separately.

Verifying Identities

These formulas are also essential for verifying trigonometric identities. When you need to prove that two expressions are equal, expanding sums or differences using these formulas often reveals the path to the solution.

Notation and Terminology

Term	Meaning	Example
Sum formula	Formula for $f(A + B)$	$\sin(A + B) = \sin A \cos B + \cos A \sin B$
Difference formula	Formula for $f(A - B)$	$\cos(A - B) = \cos A \cos B + \sin A \sin B$
Cofunction	Trig functions related by complementary angles	$\sin \theta = \cos(90° - \theta)$
Exact value	A value expressed with radicals, not decimals	$\sin 75° = \frac{\sqrt{6} + \sqrt{2}}{4}$

Examples

Key Properties and Rules

The Six Formulas

Sum Formulas: $$\sin(A + B) = \sin A \cos B + \cos A \sin B$$ $$\cos(A + B) = \cos A \cos B - \sin A \sin B$$ $$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

Difference Formulas: $$\sin(A - B) = \sin A \cos B - \cos A \sin B$$ $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$ $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Memory Aids

• Sine formulas have a mix of sine and cosine: “sine-cosine, cosine-sine”
• Cosine formulas keep functions together: “cosine-cosine, sine-sine”
• The sign in the middle of sine matches the operation (sum has plus, difference has minus)
• The sign in the middle of cosine is opposite the operation (sum has minus, difference has plus)

Common Mistakes to Avoid

• Forgetting to use the formula at all: $\sin(45° + 30°) \neq \sin 45° + \sin 30°$. Always use the sum formula.
• Mixing up the signs: Remember that cosine formulas have the opposite sign from what you might expect.
• Forgetting a term: The sine formula has two terms ($\sin A \cos B + \cos A \sin B$). Do not leave one out.
• Wrong angle in the denominator: For tangent, the denominator has the product $\tan A \tan B$, not a sum.

Derived Identities

From the sum and difference formulas, you can derive many other identities:

Cofunction identities: $$\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$$ $$\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$$

Sum-to-product formulas (like the one verified in Example 5): $$\sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$$ $$\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$$

Real-World Applications

Phase Shifts in Waves

When physicists and engineers describe waves, they often write them as $A\sin(\omega t + \phi)$, where $\phi$ is a phase shift. Using the sum formula, this can be rewritten as: $$A\sin(\omega t + \phi) = A(\sin \omega t \cos \phi + \cos \omega t \sin \phi)$$ $$= (A\cos\phi)\sin\omega t + (A\sin\phi)\cos\omega t$$

This shows that any phase-shifted sine wave can be expressed as a combination of a sine and cosine wave, which is useful for analyzing circuits and signals.

Combining Sound Waves (Interference)

When two sound waves of slightly different frequencies combine, you get “beats.” The mathematical analysis uses sum-to-product formulas derived from the sum and difference identities. If you hear two notes close in pitch, the resulting volume fluctuation is described by: $$\sin(2\pi f_1 t) + \sin(2\pi f_2 t) = 2\sin\left(2\pi\frac{f_1+f_2}{2}t\right)\cos\left(2\pi\frac{f_1-f_2}{2}t\right)$$

The cosine factor creates the “beating” effect you hear when tuning a musical instrument.

Rotation Matrices in Graphics

In computer graphics and game development, rotating a point $(x, y)$ by an angle $\theta$ uses the formulas: $$x’ = x\cos\theta - y\sin\theta$$ $$y’ = x\sin\theta + y\cos\theta$$

These are exactly the sum formulas in disguise! The new coordinates come from treating the original coordinates as sine and cosine values and applying the rotation.

AC Circuit Analysis

In electrical engineering, alternating current (AC) circuits involve voltages and currents that vary sinusoidally. When analyzing circuits with components that shift the phase between voltage and current, the sum and difference formulas are essential for calculating power and understanding interference between signals.

Self-Test Problems

Summary

• The naive approach fails: $\sin(A + B) \neq \sin A + \sin B$. Trig functions do not distribute over addition.

• Sum formulas tell you how to find trig values for $(A + B)$:

  • $\sin(A + B) = \sin A \cos B + \cos A \sin B$
  • $\cos(A + B) = \cos A \cos B - \sin A \sin B$
  • $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

• Difference formulas are the same but with flipped signs in the middle:

  • $\sin(A - B) = \sin A \cos B - \cos A \sin B$
  • $\cos(A - B) = \cos A \cos B + \sin A \sin B$
  • $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

• Finding exact values: Express non-standard angles as sums or differences of $30°$, $45°$, or $60°$, then apply the formulas. For example, $15° = 45° - 30°$ and $75° = 45° + 30°$.

• Simplifying expressions: The formulas can reduce expressions like $\sin(x + \frac{\pi}{2})$ to simpler forms (in this case, $\cos x$).

• Verifying identities: Many trig identities can be proven by expanding sums and differences using these formulas.

• Real-world applications include analyzing phase shifts in waves, understanding sound interference, performing rotations in computer graphics, and calculating AC circuit behavior.

The sum and difference formulas are among the most frequently used tools in trigonometry. They open up a world of exact values beyond the standard unit circle angles and provide the foundation for many other identities you will encounter. Master these six formulas, and you will have a powerful toolkit for solving a wide variety of problems.