Why Angle Addition Formulas Matter
Knowing the sine and cosine of two angles separately does not, by itself, tell us immediately how to calculate the sine or cosine of their sum. Angle addition formulas provide that connection. They express the trigonometric functions of a combined angle using the functions of the original angles.
The formulas are not arbitrary rules to memorize. They follow from what happens to coordinates when a point is rotated. We will use the unit circle and rotations to derive them, then use the quotient definition of tangent to obtain the corresponding tangent formulas. Throughout, the angles in a formula must use the same unit of measurement.
Coordinates on the Unit Circle
The unit circle consists of the points \((x,y)\) satisfying \(x^2+y^2=1\). A point on this circle at angle \(\theta\), measured from the positive \(x\)-axis, has coordinates \((\cos\theta,\sin\theta)\). Thus its horizontal coordinate is the cosine of its angle, and its vertical coordinate is the sine.
A rotation through an angle \(\theta\) sends the point \((1,0)\) to \((\cos\theta,\sin\theta)\). It also sends \((0,1)\) to \((-\sin\theta,\cos\theta)\): the second unit vector is the first rotated by a further quarter-turn, so its rotated coordinates are perpendicular to the first vector and point in the corresponding direction.
Every vector \((x,y)\) can be written as \(x(1,0)+y(0,1)\). Because a rotation preserves addition and scalar multiplication of vectors, its effect on an arbitrary point is therefore
This coordinate rule is the key to the derivation: rotating through \(\alpha\), then through \(\beta\), has the same result as rotating through \(\alpha+\beta\). The formulas follow by computing those same coordinates in two ways.
Deriving the Sine and Cosine Sum Formulas
Start at \((1,0)\) and rotate through \(\alpha\). The point becomes \((\cos\alpha,\sin\alpha)\). Now rotate that point through \(\beta\), using the coordinate rule above. The resulting coordinates are
But the two successive rotations together are a rotation through \(\alpha+\beta\). The final point must therefore also have coordinates \((\cos(\alpha+\beta),\sin(\alpha+\beta))\). Equating its horizontal and vertical coordinates with the expressions above proves the formulas.
The order of the terms is worth noticing. In the cosine formula, the product of the two sines is subtracted. In the sine formula, the two mixed products are added. These signs come directly from the coordinate rule for a rotation, rather than from a choice of memorization convention.
Worked Example: Rotating a Unit-Circle Point Twice
Let the starting point be \((1,0)\). After a rotation through \(\alpha\), its coordinates are \((\cos\alpha,\sin\alpha)\). Applying the rotation through \(\beta\) gives the following horizontal coordinate:
The vertical coordinate after the second rotation is
The total rotation is through \(\alpha+\beta\), so the same point has coordinates \((\cos(\alpha+\beta),\sin(\alpha+\beta))\). Matching horizontal coordinates gives the cosine sum formula; matching vertical coordinates gives the sine sum formula. As a check, the computed coordinates remain on the unit circle:
The last equality uses the Pythagorean identity. Thus the coordinate calculation is consistent with the point lying on the unit circle.
Deriving the Difference Formulas
To obtain a difference, write \(\alpha-\beta\) as \(\alpha+(-\beta)\), and use the sum formulas. The evenness of cosine and oddness of sine give \(\cos(-\beta)=\cos\beta\) and \(\sin(-\beta)=-\sin\beta\). Substituting these into the cosine sum formula yields
Using the sine sum formula in the same way gives
Worked Example: Simplifying a Difference of Cosines
Use the sum and difference formulas to simplify \(\cos(\alpha-\beta)-\cos(\alpha+\beta)\). First expand each cosine:
Distribute the subtraction across the second pair of parentheses:
Therefore,
For a check, add the two expanded expressions instead: the sine-product terms cancel, leaving \(2\cos\alpha\cos\beta\). The signs in the expansions therefore produce the expected cancellation in one combination and reinforcement in the other.
Tangent Formulas and Their Conditions
The tangent function is defined by \(\tan\theta=\sin\theta/\cos\theta\) wherever \(\cos\theta\ne 0\). To find \(\tan(\alpha+\beta)\), divide the sine sum formula by the cosine sum formula. This requires \(\cos(\alpha+\beta)\ne 0\); for the expression using \(\tan\alpha\) and \(\tan\beta\), we also require \(\cos\alpha\ne0\) and \(\cos\beta\ne0\).
The second equality follows by dividing the numerator and denominator of the first fraction by \(\cos\alpha\cos\beta\). The denominator in the resulting expression must be nonzero. Since \(\cos\alpha\cos\beta\ne0\), this is equivalent to \(\cos(\alpha+\beta)\ne0\).
Applying the same reasoning to \(\alpha-\beta\) gives the difference formula, under the corresponding conditions:
Here \(\tan\alpha\) and \(\tan\beta\) must be defined, and \(1+\tan\alpha\tan\beta\ne0\). With those individual tangents defined, the latter condition is equivalent to \(\cos(\alpha-\beta)\ne0\).
Worked Example: Applying the Tangent Sum Formula
Suppose \(\tan\alpha=\frac{2}{3}\) and \(\tan\beta=\frac{1}{4}\), with \(\cos\alpha\ne0\) and \(\cos\beta\ne0\). The denominator condition is satisfied because
Therefore the tangent of the sum is defined, and
Check the quotient directly from the sine and cosine sum formulas. Factoring \(\cos\alpha\cos\beta\) from each shows that their quotient is
The common factor is nonzero by assumption, and the remaining denominator is \(\frac{5}{6}\), also nonzero. This confirms both the calculation and the conditions for using the formula.
A Common Sign Error
A frequent mistake is to put a plus sign between the two products in the cosine sum formula. That error gives \(\cos(\alpha+\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\), which is actually the cosine difference formula.
A simple check exposes the mistake. Set \(\alpha=\beta=\frac{\pi}{2}\). The incorrect formula would give
But the left side is \(\cos\pi=-1\). The correct sum formula gives
as required. A quick check with a simple angle can help catch a sign error, but the derivation from rotations explains why the sign must be negative.
Why the Conditions Matter
The sine and cosine addition and subtraction formulas hold for all real angles \(\alpha\) and \(\beta\). Tangent formulas are different: they are quotients, so they apply only when the relevant cosine denominators are nonzero. In particular, having both \(\tan\alpha\) and \(\tan\beta\) defined does not guarantee that \(\tan(\alpha+\beta)\) is defined.
For example, if \(1-\tan\alpha\tan\beta=0\), then the cosine sum formula gives \(\cos(\alpha+\beta)=0\), provided the individual tangents are defined. The tangent of the sum is then undefined, so the quotient formula cannot be used to assign it a finite value. Always check the denominator before using a tangent addition or subtraction formula.
Check Your Understanding
Use the rotation derivation and the formulas above to answer these questions.
- What are the coordinates of a point \((x,y)\) after a rotation through \(\theta\)?
- Why does rotating through \(\alpha\) and then \(\beta\) lead to the angle \(\alpha+\beta\)?
- Derive \(\sin(\alpha-\beta)\) by replacing \(\beta\) with \(-\beta\) in a sum formula.
- State the condition needed to use the tangent sum formula in terms of \(\tan\alpha\) and \(\tan\beta\).
- Can the sine and cosine sum formulas be used when \(\cos\alpha=0\)? What about the tangent sum formula written using \(\tan\alpha\)?
