An addition formula applied to itself

Replace $b=a$ in addition formulas:

$$\sin2a=2\sin a\cos a,$$

$$\cos2a=\cos^2a-\sin^2a.$$

The fundamental identity allows two other forms for the cosine:

$$\cos2a=1-2\sin^2a=2\cos^2a-1.$$

The choice depends on the data available. If we only know the sine, the first alternative expression avoids calculating the cosine and choosing its sign.

If tangents exist:

$$\tan2a=\frac{2\tan a}{1-\tan^2a}.$$

The form with $\tan a$ requires $\cos a\ne0$ and $1-\tan^2a\ne0$; does not cover all cases where $\tan2a$ exists.

Demonstration of triple formulas

Write $3a=2a+a$. For the sine:

$$\begin{aligned} \sin3a&=\sin2a\cos a+\cos2a\sin a\\ &=2\sin a\cos^2a+(1-2\sin^2a)\sin a\\ &=2\sin a(1-\sin^2a)+\sin a-2\sin^3a\\ &=3\sin a-4\sin^3a. \end{aligned}$$

In the same way, expanding the cosine of the sum and substituting $\sin^2a=1-\cos^2a$:

$$\cos3a=4\cos^3a-3\cos a.$$

Dividing sine by cosine and simplifying, with $t=\tan a$:

$$\tan3a=\frac{3t-t^3}{1-3t^2},$$

where $\tan a$ exists and $1-3t^2\ne0$. The zero denominator corresponds to an interruption in the expression.

Example with known reasons

If $\sin a=3/5$ and $\cos a=4/5$:

$$\sin2a=\frac{24}{25},\qquad\cos2a=\frac7{25},$$

$$\sin3a=\frac{117}{125},\qquad\cos3a=-\frac{44}{125}.$$

The positive sine and negative cosine show that the image of $3a$ is in the second quadrant. The sign may change when multiplying the angle.

In everyday life

  • Throws. The range of a throw is proportional to $\operatorname{sen} 2\theta$; That’s why $30^\circ$ and $60^\circ$ arrive at the same place and $45^\circ$ goes further.
  • Electrical power. The power in a resistor is proportional to $\operatorname{sen}^2\omega t = \frac{1 - \cos 2\omega t}{2}$: it oscillates at twice the frequency of the voltage, 120 times per second.
  • Triple angles. Dividing an angle into three (trisection) leads to a 3rd degree equation, the same one that appears in $\cos 3a$ — that’s why it can’t be done with a ruler and compass.
Double arc in sport: range is proportional to $\operatorname{sen}2\theta$.

Exercises with resolution

1. Express $\cos4a$ only in terms of $c=\cos a$.

Check resolution 1

$\cos4a=2\cos^22a-1=2(2c^2-1)^2-1=8c^4-8c^2+1$.

2. If $\tan a=1/2$, calculate $\tan2a$.

Check resolution 2

$\tan2a=2(1/2)/(1-1/4)=4/3$. The denominators involved are nonzero.

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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