Pythagoras in the circle

The point $P=(\cos x,\sin x)$ is one unit away from the origin. The projections on the axes form a right-angled triangle with sides $|\cos x|$ and $|\sin x|$. Applying Pythagoras, we obtain:

$$\sin^2x+\cos^2x=1.$$

On the axes, equality is verified directly: one coordinate is zero and the other has modulus 1. Therefore, the identity holds for every real $x$.

Bring relationships together

The definitions and previous identity provide:

$$\tan x=\frac{\sin x}{\cos x},\quad \cot x=\frac{\cos x}{\sin x},$$

$$\sec x=\frac1{\cos x},\quad\csc x=\frac1{\sin x}.$$

Dividing the fundamental identity by $\cos^2x$ requires $\cos x\ne0$ and gives:

$$1+\tan^2x=\sec^2x.$$

Dividing by $\sin^2x$, with $\sin x\ne0$, gives:

$$1+\cot^2x=\csc^2x.$$

When sine and cosine are not zero, $\tan x\cot x=1$ is still valid. These results do not dispense with the conditions of the denominators.

Example: retrieve the other functions

Assume $\sin x=5/13$ and $x$ in the second quadrant. Then:

$$\cos^2x=1-\frac{25}{169}=\frac{144}{169}.$$

The cosine is negative in this quadrant, therefore $\cos x=-12/13$. It follows that:

$$\tan x=-\frac5{12},\quad\cot x=-\frac{12}{5},\quad \sec x=-\frac{13}{12},\quad\csc x=\frac{13}{5}.$$

The square root provides two sign possibilities; the quadrant is the data that allows you to choose.

A warning about notation

$\sin^2x$ means $(\sin x)^2$. $\sin(x^2)$ applies sine to the square of the input and is a different function. Likewise, an identity between squares does not allow directly equating the values ​​without considering the signs.

In everyday life

  • Inclined plane. The components $P\operatorname{sen}\alpha$ and $P\cos\alpha$ of the weight on a ramp recompose the weight by the fundamental relationship.
  • Electrical energy. Active, reactive and apparent power form a right-angled triangle: $S^2 = P^2 + Q^2$, with $P = S\cos\varphi$ and $Q = S\operatorname{sen}\varphi$.
  • Retrieve data. If a sensor only measures the cosine of an angle, the fundamental relationship returns the sine — only the signal is missing, which comes from the quadrant.
Sine and cosine of the same angle recompose the diagonal of the screen.

Exercises with resolution

1. If $\tan x=3/4$, calculate $\sec^2x$.

Check resolution 1

$\sec^2x=1+9/16=25/16$. The square has been determined; the sign of the secant depends on the quadrant.

2. If $\cos x=-3/5$ and $x$ are in the third quadrant, calculate the sine.

Check resolution 2

$\sin^2x=1-9/25=16/25$. In the third quadrant, $\sin x=-4/5$.

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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