A point and its reflections

Choose $0<\alpha<\pi/2$, whose image is $(\cos\alpha,\sin\alpha)$. A reflection on the vertical axis changes the sign of the first coordinate; a reflection on the horizontal axis exchanges that of the second; symmetry about the origin exchanges both.

These three operations generate the points of $\pi-\alpha$, $\pi+\alpha$ and $2\pi-\alpha$. Logo:

Angle Sine Cosine Tangent
$\pi-\alpha$ $\sin\alpha$ $-\cos\alpha$ $-\tan\alpha$
$\pi+\alpha$ $-\sin\alpha$ $-\cos\alpha$ $\tan\alpha$
$2\pi-\alpha$ $-\sin\alpha$ $\cos\alpha$ $-\tan\alpha$

Demonstration for the second quadrant

The reflection of $P=(u,v)$ on the vertical axis is $P'=(-u,v)$. The height remains the same, but the horizontal position becomes negative. Since $P'$ corresponds to angle $\pi-\alpha$:

$$\sin(\pi-\alpha)=\sin\alpha,\qquad \cos(\pi-\alpha)=-\cos\alpha.$$

Dividing the coordinates, $\tan(\pi-\alpha)=-\tan\alpha$. The other rows of the table are demonstrated by the corresponding reflections.

The other three functions

Cotangent has the same sign as tangent; secant has the same sign as cosine; cosecant, the same sign as the sine. Taking the reciprocals of the table entries provides their values, respecting the points where the denominators cannot be zero.

Three examples

$$\sin\frac{5\pi}{6}=\sin\frac\pi6=\frac12,$$

$$\cos\frac{5\pi}{4}=-\cos\frac\pi4=-\frac{\sqrt2}{2},$$

$$\tan\frac{11\pi}{6}=-\tan\frac\pi6=-\frac{\sqrt3}{3}.$$

First we identify the quadrant, then the reference angle and finally the sign. If the input has multiple loops, we remove multiples of $2\pi$ before these steps.

In everyday life

  • Calculators and computers. Every calculator reduces the arc to the 1st quadrant before calculating, which allows you to save a small table.
  • Navigation. A heading in the 3rd quadrant, like $225^\circ$, has two negative components: the boat goes west and south.
  • Games. The direction of a character facing $150^\circ$ is obtained from the direction of $30^\circ$ by changing the sign of the horizontal component.
Directions in any quadrant: the cosine and sine signs give the direction of displacement.

Exercises with resolution

1. Calculate $\cos(7\pi/6)$ and $\sin(7\pi/4)$.

Check resolution 1

$7\pi/6=\pi+\pi/6$, then the cosine is $-\sqrt3/2$. As $7\pi/4=2\pi-\pi/4$, the sine is $-\sqrt2/2$.

2. Calculate $\sec(2\pi/3)$.

Check resolution 2

$\cos(2\pi/3)=-1/2$. Its reciprocal is $-2$.

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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