Reduce the table further

In a right-angled triangle, the acute angles are complementary. The side opposite to one is adjacent to the other. Like this:

$$\sin\left(\frac\pi2-x\right)=\cos x,\qquad \cos\left(\frac\pi2-x\right)=\sin x.$$

For $\pi/4\le x\le\pi/2$, the complement $\pi/2-x$ belongs to $[0,\pi/4]$. In this way, values ​​in the second half of the first quadrant can be calculated from those in the first half, exchanging sine for cosine.

Also:

$$\tan\left(\frac\pi2-x\right)=\cot x,$$

$$\sec\left(\frac\pi2-x\right)=\csc x,$$

in their respective domains. The geometric interpretation by exchanging the legs demonstrates the relationships for acute angles; reading the cycle coordinates extends the identities to the other allowed values.

Identities for any lap

The symmetries of the cycle provide, for every real $x$:

$$\sin(\pi-x)=\sin x,\quad\cos(\pi-x)=-\cos x,$$

$$\sin(x+\pi)=-\sin x,\quad\cos(x+\pi)=-\cos x.$$

Removing multiples of $2\pi$ reduces any entry to the first turn. These formulas then allow you to work in the first quadrant without losing the signals.

Even and odd functions

The domain of an even or odd function must be symmetric about zero. The function is even if $f(-x)=f(x)$ and odd if $f(-x)=-f(x)$.

Reflection on the horizontal axis preserves the cosine and inverts the sine. Therefore, cosine and secant are pairs; sine, tangent, cotangent and cosecant are odd. Assertions hold where functions exist.

An example of simplification

Using complement and parity:

$$\begin{aligned} \sin(-x)+\cos\left(\frac\pi2-x\right) &=-\sin x+\sin x=0. \end{aligned}$$

This equality is not just a coincidence for a notable angle: it holds for every real $x$.

In everyday life

  • Noise cancellation. The headphones emit the noise wave with the signal changed; the sine being odd explains why this cancels it out.
  • Tables up to $45^\circ$. The printed tables used the sine ↔ cosine exchange of complements to bring only half of the values.
  • Sun and zenith. The height of the Sun above the horizon and the distance to the zenith are complementary: the sine of one is the cosine of the other.
Odd sine: the wave with the changed sign cancels the original.

Exercises with resolution

1. If $\cos15°=a$, express $\sin75°$ by $a$.

Check resolution 1

Such as $75°=90°-15°$, $\sin75°=\cos15°=a$.

2. Simplify $\cos(-x)-\cos x+\tan(-x)+\tan x$.

Check resolution 2

Cosine parity cancels out the first two terms; the tangent impairment cancels out the others. The result is 0, in domain $\cos x\ne0$.

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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