An algebra for angles: split, combine, double, and merge.
Up to now an angle has been a single input and sin θ a single output. This lesson treats sines and cosines as quantities you can do algebra with: add two angles, double one, halve one, or fold two separate waves into a single wave. The engine is one fact about a rotation — the cosine of a difference, cos(α − β) = cos α cos β + sin α sin β — and everything else in the chapter is squeezed out of it. We will check each rule with real numbers so it never feels like a magic spell: cos 75° really does equal cos(45° + 30°), and 3 sin x + 4 cos x really is one wave of height 5. By the end you will hold a small toolbox of identities — sum, difference, double-angle, power-reduction, and the auxiliary-angle merge — that turns a tangle of trig into something you can simplify on sight. It rests on the unit circle (25.3) and the wave graphs (25.4), and it powers the harmonic models of 25.6.
Here is the one fact the whole chapter rests on. Take two points on the unit circle, A at angle α and B at angle β. From 25.3, their coordinates are A(cos α, sin α) and B(cos β, sin β). The angle between the two radii is α − β. Computing the distance AB two ways — once with the distance formula, once with the law of cosines on the triangle OAB (both radii equal 1) — and setting the two expressions equal makes every term cancel except one clean relation:
cos(α + β) = cos α cos β − sin α sin β
cos(α − β) = cos α cos β + sin α sin β
The cosine of a combined angle is not the sum of the cosines. It mixes both functions, and the sign of the last term is the opposite of the sign inside the angle: a plus inside gives a minus term, a minus inside gives a plus term.
75° is not a special angle, but 75° = 45° + 30°, and both of those are. Using the plus form:
cos 75° = cos 45° cos 30° − sin 45° sin 30° = √22·√32 − √22·12 = √6 − √24 ≈ 0.259.
A calculator gives cos 75° ≈ 0.259 — the formula and the direct value agree exactly.
Once you own the cosine rule, the sine rule comes free. A sine is just a cosine of the complementary angle: sin θ = cos(90° − θ). So sin(α + β) = cos(90° − α − β) = cos((90° − α) − β), and expanding that difference with the rule above untangles into a clean pattern. The tangent rule then follows by dividing the sine formula by the cosine formula.
sin(α ± β) = sin α cos β ± cos α sin β
tan(α ± β) = tan α ± tan β1 ∓ tan α tan β
For sine the sign inside the angle is the same sign you use in the formula (a useful contrast with cosine, where it flips). For tangent, the top sign matches the angle and the bottom sign is the opposite.
sin 75° = sin 45° cos 30° + cos 45° sin 30° = √22·√32 + √22·12 = √6 + √24 ≈ 0.966.
And tan 75° = tan(45° + 30°) = 1 + 1/√31 − 1/√3 = 2 + √3 ≈ 3.732 — exactly what a calculator returns.
The single most common trig error is writing sin(α + β) = sin α + sin β. It is false. Take α = β = 30°: the left side is sin 60° ≈ 0.866, while sin 30° + sin 30° = 1. The correct rule mixes a sine with a cosine — never just adds two sines.
What happens when the two angles are the same? Set β = α in the sum formulas and the right-hand sides collapse. This is the cheapest, most-used family of identities: the double-angle formulas.
sin 2α = 2 sin α cos α
cos 2α = cos2α − sin2α = 1 − 2 sin2α = 2 cos2α − 1
tan 2α = 2 tan α1 − tan2α
The three forms of cos 2α are all equal — you reach the second and third by swapping sin2α ↔ 1 − cos2α using the Pythagorean identity sin2α + cos2α = 1. Pick whichever leaves only the function you want.
Setting α = 30°, sin 60° = 2 sin 30° cos 30° = 2 · 12 · √32 = √32 ≈ 0.866 — exactly sin 60°. And cos 60° = 1 − 2 sin230° = 1 − 2·14 = ½.
Run the cosine double-angle backward. From cos 2α = 1 − 2 sin2α, solve for sin2α; from cos 2α = 2 cos2α − 1, solve for cos2α. The result trades a squared trig function — awkward to integrate or graph — for a plain cosine of the doubled angle.
sin2α = 1 − cos 2α2 cos2α = 1 + cos 2α2
Replacing α with θ/2 turns these into the half-angle formulas, sin2(θ/2) = (1 − cos θ)/2. They are the trick of choice for integrating sin2x in calculus and for finding exact values like sin 15°.
sin215° = 1 − cos 30°2 = 1 − √3/22 = 2 − √34 ≈ 0.067. Since 15° is in Ⅰ, sin 15° = +√(2 − √3)/2 ≈ 0.259.
Here is the identity that pays for the whole chapter in physics. A signal made of a sine and a cosine of the same frequency — say 3 sin x + 4 cos x — looks like two separate vibrations. But the sine sum formula runs backward to fold them into a single sine wave. Write a sin x + b cos x and ask: is there an amplitude R and a phase shift φ so that R sin(x + φ) = a sin x + b cos x for all x? Expanding R sin(x + φ) = (R cos φ) sin x + (R sin φ) cos x, match coefficients: R cos φ = a and R sin φ = b. Squaring and adding gives R, and dividing gives φ.
a sin x + b cos x = R sin(x + φ), R = √(a2 + b2), tan φ = ba
Two waves of the same frequency always add up to one wave of that frequency — taller (height R) and shifted (by φ). Nothing new is born; the pair just re-expresses one sinusoid.
R = √(32 + 42) = √25 = 5, and tan φ = 4/3, so φ = 53.13° (≈ 0.927 rad). Therefore 3 sin x + 4 cos x = 5 sin(x + 53.13°). Check at x = 0: the left side is 4; the right is 5 sin 53.13° = 5·0.8 = 4. They agree.
One rule generates the rest. Keep this sheet in your head; everything else is a substitution away.
| Name | Identity |
|---|---|
| Cosine sum / difference | cos(α ± β) = cos α cos β ∓ sin α sin β |
| Sine sum / difference | sin(α ± β) = sin α cos β ± cos α sin β |
| Tangent sum / difference | tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β) |
| Double angle | sin 2α = 2 sin α cos α; cos 2α = 1 − 2 sin2α |
| Power reduction | sin2α = (1 − cos 2α)/2; cos2α = (1 + cos 2α)/2 |
| Auxiliary angle | a sin x + b cos x = R sin(x + φ), R = √(a2+b2) |
Use the difference formula to find cos 15° exactly, writing 15° = 45° − 30°.
cos 15° = cos 45° cos 30° + sin 45° sin 30° = √22·√32 + √22·12 = (√6 + √2)/4 ≈ 0.966. (The cosine difference takes a plus middle sign.)
Show that sin 2α = 2 sin α cos α follows from the sine sum formula. Then evaluate sin 90° using α = 45°.
Put β = α in sin(α + β) = sin α cos β + cos α sin β: sin 2α = sin α cos α + cos α sin α = 2 sin α cos α. With α = 45°: sin 90° = 2·(√2/2)·(√2/2) = 2·½ = 1. ✓
A student claims cos(α + β) = cos α + cos β. Disprove it with α = β = 60°.
Left side: cos 120° = −½. Right side: cos 60° + cos 60° = ½ + ½ = 1. Since −½ ≠ 1, the claim is false — cosine does not distribute over addition.
Use power reduction to rewrite cos230° and confirm it equals ¾.
cos230° = 1 + cos 60°2 = 1 + ½2 = ¾. Directly, cos 30° = √3/2, so cos230° = 3/4. ✓
Merge √3 sin x + cos x into a single sine R sin(x + φ). Give R and φ in degrees.
R = √((√3)2 + 12) = √4 = 2; tan φ = 1/√3, so φ = 30°. Hence √3 sin x + cos x = 2 sin(x + 30°). Check x = 0: left = 1, right = 2 sin 30° = 1. ✓
Find the maximum value of 5 sin x + 12 cos x and the smallest positive x (in degrees) where it occurs.
R = √(52 + 122) = √169 = 13, and tan φ = 12/5 so φ ≈ 67.38°. The expression is 13 sin(x + 67.38°), whose maximum is 13, reached when x + 67.38° = 90°, i.e. x ≈ 22.62°.
Six questions to lock it in. Tap the answer you think is right.
This lesson covers the sum and difference identities, the double-angle and half-angle (power-reduction) formulas, and the linear-combination (auxiliary-angle) form a sin x + b cos x = R sin(x + φ). It aligns with the Common Core functions standards HSF-TF.C.9 (prove the addition and subtraction formulas and use them) and the trigonometric-functions cluster HSF-TF.C.8 (the Pythagorean identity), and prepares students for harmonic models in 25.6. The widgets verify each identity numerically, so students see that an "identity" is a statement true for every input, not just a manipulation rule.