Introduction
Study trigonometry until you open a table of identities and you sigh. There are the addition formulas that give the sine and cosine of a sum of two angles, the double-angle formulas that overlap the same angle twice, the half-angle formulas that fold an angle in half. At a glance there are six or seven separate formulas to memorize.
$$\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$$$$\cos 2\alpha = \cos^2\alpha - \sin^2\alpha, \qquad \sin^2\tfrac{\theta}{2} = \frac{1-\cos\theta}{2}$$Here \(\alpha\) (alpha), \(\beta\) (beta), and \(\theta\) (theta) are all Greek letters standing for angles. Many people memorize these formulas one by one as if they were strangers. But they are in fact one family. Hold on to a single seed formula and everything else grows out of it.
The seed is the one at the top, \(\cos(\alpha-\beta)\), the cosine of the difference of two angles. Once you understand where this one comes from, a single path opens up: overlap the two angles to get the double-angle formula, then solve that double-angle formula backwards to get the half-angle formula. In this post we follow that family tree in pictures.
1. Planting the seed — a cosine from the angle between two arrows
First move the stage to the unit circle (the circle of radius \(1\)). Consider two arrows reaching from the origin to points on the circle. One is the arrow \(\vec{A}\) turned by angle \(\alpha\) from the horizontal axis, the other the arrow \(\vec{B}\) turned by angle \(\beta\). Using the definition of the trig functions directly (cosine of the angle = horizontal coordinate of the point, sine = vertical coordinate), the coordinates of the two arrows are:
$$\vec{A} = (\cos\alpha,\ \sin\alpha), \qquad \vec{B} = (\cos\beta,\ \sin\beta)$$Now compute the dot product of the two arrows (the multiplication that measures how much they face the same direction) in two different ways. Measuring the dot product two ways must give the same value, so setting the two values side by side makes a formula pop out.
Way 1 — by coordinate components. The dot product is the product of the horizontals plus the product of the verticals.
$$\vec{A}\cdot\vec{B} = \cos\alpha\cos\beta + \sin\alpha\sin\beta$$Way 2 — by the angle between them. The dot product is also "the product of the two arrow lengths, times the cosine of the angle between them." Both arrows have length \(1\), and the angle they spread apart is the difference of \(\alpha\) and \(\beta\), namely \(\alpha-\beta\). So
$$\vec{A}\cdot\vec{B} = 1\cdot 1\cdot\cos(\alpha-\beta) = \cos(\alpha-\beta)$$We measured the same dot product twice, so the two results must be equal. Joining them with an equals sign completes the seed formula.
$$\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta$$This is not something memorized but the result of measuring the same thing twice and matching. (Why the dot product is "length times length times the cosine of the angle between," seen as a shadow, is covered separately in Why the Dot Product Produces a Cosine.)
2. The addition family grows from the seed
Holding a single seed, we grow the rest of the addition formulas by twisting the angles just slightly.
Difference into sum. Put \(-\beta\) in the place of \(\beta\). Cosine is unchanged by a negative angle (\(\cos(-\beta)=\cos\beta\)), while sine flips sign (\(\sin(-\beta)=-\sin\beta\)). Substituting into the seed formula:
$$\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$$Only one sign changed. (Why only sine flips sign under a negative angle is covered in Flip an Angle Negative and Why Only Sine Changes Sign.)
From cosine to sine. Use the relation that a sine equals the cosine of the complementary angle (the angle that sums to \(90°\) with it), that is, \(\sin\theta = \cos(90°-\theta)\). Rewrite \(\sin(\alpha+\beta)\) as a cosine with this relation, then apply the seed formula again:
$$\sin(\alpha+\beta) = \cos\big(90°-(\alpha+\beta)\big) = \cos\big((90°-\alpha)-\beta\big)$$The right side has the seed formula's shape (the cosine of a difference of two angles). Expanding, since \(\cos(90°-\alpha)=\sin\alpha\) and \(\sin(90°-\alpha)=\cos\alpha\),
$$\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$Layering onto one seed only the rules we already know — negative angle and complementary angle — gives all four lines of the addition formulas. Nothing new was memorized.
3. Overlap the two angles — the double-angle formulas
Now the prettiest part. In the addition formula, make the two angles the same. Put \(\alpha\) in the place of \(\beta\), that is, overlap the two arrows. Then \(\alpha+\beta\) becomes \(2\alpha\), and the formula folds on its own into the double-angle formula.
$$\cos 2\alpha = \cos\alpha\cos\alpha - \sin\alpha\sin\alpha = \cos^2\alpha - \sin^2\alpha$$$$\sin 2\alpha = \sin\alpha\cos\alpha + \cos\alpha\sin\alpha = 2\sin\alpha\cos\alpha$$The double-angle formulas are not a separate invention but the special case of the addition formulas with the two angles overlapped.
Slip in just the Pythagorean identity \(\sin^2\alpha+\cos^2\alpha=1\) and the cosine double-angle formula splits into three faces. Replacing \(\sin^2\alpha = 1-\cos^2\alpha\) makes sine vanish; replacing \(\cos^2\alpha=1-\sin^2\alpha\) makes cosine vanish.
$$\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha$$The three expressions merely look different; they are always the same value. (How many formulas spring from this single identity is covered in Why Sine Squared Plus Cosine Squared Is Always 1.)
4. Solve it backwards — the half-angle formulas
Finally, take the last of the three faces just found and solve it backwards. Rearranging \(\cos 2\alpha = 1-2\sin^2\alpha\) for \(\sin^2\alpha\):
$$\sin^2\alpha = \frac{1-\cos 2\alpha}{2}$$Now read the angle on a half-scale. Call all of \(2\alpha\) by the name \(\theta\), so \(\alpha\) becomes \(\theta/2\), and the formula above is exactly the half-angle formula.
$$\sin^2\frac{\theta}{2} = \frac{1-\cos\theta}{2}, \qquad \cos^2\frac{\theta}{2} = \frac{1+\cos\theta}{2}$$(The cosine side comes from solving \(\cos 2\alpha = 2\cos^2\alpha - 1\) the same way.) Why a \(\pm\) sign appears when you take the square root to get \(\sin(\theta/2)\) itself also becomes clear here. Undoing a square erases the sign information, so you have to read which quadrant \(\theta/2\) lies in and set the sign separately.
The half-angle formula, too, was not a new formula but the double-angle formula read backwards.
5. One picture — the family tree of the formulas
Stringing the whole flow into one line:
- Seed: measure the dot product of two arrows two ways to get \(\cos(\alpha-\beta) = \cos\alpha\cos\beta+\sin\alpha\sin\beta\).
- Addition formulas: layer the negative angle (\(\beta\to-\beta\)) and complementary angle (\(\sin\) as the cosine of the complement) onto the seed for the other three lines.
- Double angle: overlap the two angles in the addition formula by setting \(\beta=\alpha\).
- Half angle: solve the double-angle formula backwards and reread it on a half-scale.
A whole page of the formula table was really one seed and its variations. Instead of memorizing them one by one, just ask "how did this grow from the seed?" — and even when memory grows fuzzy, you can regrow it on the spot. The best way to reduce what you must memorize in math is not to memorize better but to notice that what looked like many things was really one.
6. See it for yourself
In the interactive below, move the angles \(\alpha\) and \(\beta\) of the two arrows with the sliders.
In addition mode, you can watch the value computed by coordinate components, \(\cos\alpha\cos\beta+\sin\alpha\sin\beta\), and the value measured by the angle between, \(\cos(\alpha-\beta)\), match all the way to the decimals. No matter how you turn the two arrows the two values never disagree — that is what it means for the seed formula to always hold.
Turn on double-angle mode and \(\beta\) overlaps onto \(\alpha\) so the two arrows become one. Now you can confirm numerically that the three faces of the cosine double-angle formula — \(\cos^2\alpha-\sin^2\alpha\), \(2\cos^2\alpha-1\), and \(1-2\sin^2\alpha\) — all gather to the same value \(\cos 2\alpha\).
Key takeaways
- A table of trig identities is not a list to memorize item by item but one seed and its variations. The seed is \(\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\), obtained by measuring the dot product of two arrows on the unit circle two ways.
- The rest of the addition formulas come out by layering onto the seed only rules you already know: the negative angle (\(\cos(-\beta)=\cos\beta\), \(\sin(-\beta)=-\sin\beta\)) and the complementary angle (\(\sin\theta=\cos(90°-\theta)\)).
- The double-angle formulas are the special case of the addition formulas with the two angles overlapped (\(\beta=\alpha\)): \(\cos 2\alpha=\cos^2\alpha-\sin^2\alpha\), \(\sin 2\alpha=2\sin\alpha\cos\alpha\). Slip in the Pythagorean identity and \(\cos 2\alpha=2\cos^2\alpha-1=1-2\sin^2\alpha\) become three faces.
- The half-angle formulas are the double-angle formulas solved backwards: \(\sin^2\tfrac{\theta}{2}=\tfrac{1-\cos\theta}{2}\), \(\cos^2\tfrac{\theta}{2}=\tfrac{1+\cos\theta}{2}\). The \(\pm\) when you take the square root is because you must set the sign by quadrant.
- The way to reduce what you memorize is not to memorize better but to notice that what looked like many things was really one.
Related reading — Why the Dot Product Produces a Cosine (the identity of the dot product that gives birth to the seed formula) · Why Sine Squared Plus Cosine Squared Is Always 1 (the identity that splits the double angle into three faces) · Flip an Angle Negative and Why Only Sine Changes Sign (the negative- and complementary-angle rules used to grow the addition formulas) · What on Earth Is the Point of Trigonometry (why the addition formula is wave synthesis)