Addition, double-angle and half-angle formulas

What is sin 75°? The first idea that comes to mind: 75° = 45° + 30°, so sin 75° = sin 45° + sin 30° ≈ 0.707 + 0.5 = 1.207. But a sine can never exceed 1! A calculator shows sin 75° ≈ 0.966. So the sine of a sum is not the sum of the sines — it has a formula of its own. In this lesson we derive that formula and obtain from it all the main formulas of trigonometry.

This lesson continues “Radian measure, basic identities and reduction formulas”: the unit circle, the identity sin²α + cos²α = 1 and the signs in the quadrants are needed at every step. The plan: first the sum and difference formulas, from them the double-angle formulas, from those power reduction and half angles, then sum-to-product, and finally the expression a·sin x + b·cos x. Each new formula takes one or two lines to get from the previous one, so what matters is not memorising them all but being able to derive them.

Sum and difference formulas

Everything starts from one formula, the cosine of a difference. We derive it by computing the distance between two points of the unit circle in two ways:

1. Take the points A(cos α, sin α) and B(cos β, sin β); the angle AOB equals α − β.
2. In coordinates: AB² = (cos α − cos β)² + (sin α − sin β)². Expanding and using sin² + cos² = 1 gives AB² = 2 − 2(cos α cos β + sin α sin β).
3. Rotate the picture back through β: B goes to (1, 0) and A to (cos(α − β), sin(α − β)), and the distance does not change. Then AB² = (cos(α − β) − 1)² + sin²(α − β) = 2 − 2cos(α − β).
4. Set the two expressions equal: cos(α − β) = cos α cos β + sin α sin β. Replacing β with −β (cos is even, sin is odd) gives the cosine of a sum.

The addition formulas contain all the reduction formulas of the previous lesson. For example, sin(π/2 + α) = sin(π/2) cos α + cos(π/2) sin α = 1 · cos α + 0 · sin α = cos α, and cos(π − α) = cos π cos α + sin π sin α = −cos α. The horse rule is just a shortcut for these calculations: since the sine and cosine of π/2, π and 3π/2 are 0 or ±1, one of the two terms always vanishes.

Double-angle formulas

Put β = α in the addition formulas: sin(α + α) = sin α cos α + cos α sin α = 2 sin α cos α and cos(α + α) = cos²α − sin²α. Substituting sin²α = 1 − cos²α or cos²α = 1 − sin²α into the cosine formula gives two more forms. Pick the form that uses the function you are given.

The double-angle formula has a neat consequence: sin α cos α = ½ sin 2α ≤ ½. For example, a right triangle with hypotenuse c has legs c sin α and c cos α, so its area is S = ½ · c sin α · c cos α = ¼ c² sin 2α. It is largest when α = 45°, i.e. when the triangle is isosceles, and then equals c²/4.

Power-reduction and half-angle formulas

Solve cos 2α = 1 − 2sin²α and cos 2α = 2cos²α − 1 for sin²α and cos²α. The squares disappear and the angle doubles, which is why these are called the power-reduction formulas. In equations and integrals they get rid of expressions such as sin²x and cos²x.

Sum-to-product and product-to-sum

Add the formulas for sin(x + y) and sin(x − y): sin(x + y) + sin(x − y) = 2 sin x cos y. Now let x + y = α and x − y = β; then x = (α + β)/2, y = (α − β)/2 and sin α + sin β = 2 sin((α + β)/2) cos((α − β)/2). The other three formulas come the same way. In product form it is easy to factor equations and cancel fractions.

a·sin x + b·cos x and exam-style transformations

What is the largest value of 3 sin x + 4 cos x? Not 3 + 4 = 7: sine and cosine do not reach their maximum at the same x. Take the factor R = √(a² + b²) out of the expression. Since (a/R)² + (b/R)² = 1, there is an angle φ with cos φ = a/R and sin φ = b/R. The bracket then becomes sin x cos φ + cos x sin φ = sin(x + φ) — this is the auxiliary angle method.

1. 1
   Find R

   Compute R = √(a² + b²).

2. 2
   Take R out of the bracket

   Write a sin x + b cos x = R · ((a/R) sin x + (b/R) cos x).

3. 3
   Choose φ

   Find the angle with cos φ = a/R and sin φ = b/R; take both signs into account.

4. 4
   Fold up and conclude

   The bracket becomes sin(x + φ). The largest value is R, the smallest −R.