Rotate a vector

Consider the horizontal and vertical unit vectors. After a rotation of $a$, they start to point, respectively, to $(\cos a,\sin a)$ and $(-\sin a,\cos a)$. The second is perpendicular to the first, in the positive direction.

The point corresponding to $b$ is the sum of $\cos b$ times the horizontal vector with $\sin b$ times the vertical. Rotation preserves this decomposition. After turning $a$:

$$\begin{aligned} (\cos(a+b),\sin(a+b)) &=\cos b(\cos a,\sin a)\\ &\quad+\sin b(-\sin a,\cos a). \end{aligned}$$

Demonstrate and assemble the formulas

Comparing the coordinates, we obtain the sum formulas. Replacing $b$ with $-b$ and using parity gives the difference:

$$\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b,$$

$$\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b.$$

The sign choices are corresponding: the top sign in the whole formula or the bottom sign in the whole formula.

Dividing sine by cosine and then numerator and denominator by $\cos a\cos b$:

$$\tan(a\pm b)=\frac{\tan a\pm\tan b}{1\mp\tan a\tan b}.$$

This writing requires that $\tan a$, $\tan b$ and the result exist. In particular, the denominator cannot be zero.

Dividing cosine by sine and using $\cot a=\cos a/\sin a$:

$$\cot(a+b)=\frac{\cot a\cot b-1}{\cot a+\cot b},$$

$$\cot(a-b)=\frac{\cot a\cot b+1}{\cot b-\cot a}.$$

Both of these forms require $\sin a\sin b\ne0$ and non-zero trailing denominators. An expression with cotangents can have a smaller domain than the quotient written directly with sines and cosines.

Example of a new value

For $75°=45°+30°$:

$$\begin{aligned} \sin75° &=\frac{\sqrt2}{2}\frac{\sqrt3}{2} +\frac{\sqrt2}{2}\frac12\\ &=\frac{\sqrt6+\sqrt2}{4}. \end{aligned}$$

The difference formula gives $\sin15°=(\sqrt6-\sqrt2)/4$. Adding angles does not mean adding their sines: $\sin(a+b)$ involves the two coordinates.

In everyday life

  • Image rotation. Rotating a photo applies the rotation matrix to each point, which is nothing more than the sine and cosine sum formulas.
  • Maps on your cell phone. Adding the compass rotation to the direction shown on the screen is adding arcs.
  • New values. With $45^\circ$ and $30^\circ$ exact values ​​for $15^\circ$ and $75^\circ$ are obtained, useful in carpentry cuts and fittings.
The rotation of a figure, point by point, by addition formulas.

Exercises with resolution

1. Calculate $\cos75°$.

Check resolution 1

$\cos(45°+30°)=\sqrt6/4-\sqrt2/4=(\sqrt6-\sqrt2)/4$.

2. Calculate $\tan15°$.

Check resolution 2

$\tan(45°-30°)=(1-1/\sqrt3)/(1+1/\sqrt3)=2-\sqrt3$.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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