Properties of sine, cosine, tangent and cotangent

Sine and cosine are basic goniometric functions.

Sine

The sine of the angle alpha is equal to the ratio of the length of the opposite branch to the length of the hypotenuse in a right triangle.

$$\sin(\alpha)=\frac{\mbox{ The length of the opposite hanger }}{\mbox{ Delka prepona }}$$

The graph of the sine function is a curve called the sine. You can see the graph in the following figure:

The sine is the periodic function that has the smallest period 2π. Other properties:

• Thedefining domain is the set of all real numbers.
• The range of values is the interval <−1,1>.
• Sine has a maximum at infinitely many points. More precisely, it has its "first" maximum at a point $x=\frac{\pi}{2}$ and, since it is a periodic function, it also has a maximum at every point $\frac{\pi}{2}+2k\pi$, where k is an integer. The value of the maximum is then 1.
• Similarly for the minimum: the sine has a minimum at the points $-\frac{\pi}{2}+2k\pi$, where k is an integer and its value is −1.
• Sine is an odd and bounded function.

Cosine

The cosine of the angle alpha is equal to the ratio of the length of the adjacent branch to the length of the hypotenuse in a right triangle. So if we calculate the cosine of the angle alpha on a calculator, we get the value of the ratio of

$$\cos(\alpha)=\frac{\mbox{ length of the adjacent hanger }}{\mbox{ length of the prefix }}.$$

The graph of the cosine function is a curve called the cosine. You can see the graph in the following figure:

Cosine is a periodic function that has the smallest period 2π. Other properties:

• Thedefining domain is the set of all real numbers.
• The range of values is the interval <−1,1>.
• Cosine has a maximum at infinitely many points. More precisely, it has its "first" maximum at a point x = 0, and since it is a periodic function, it also has a maximum at every point 2kπ, where k is an integer. The value of the maximum is then 1.
• Similarly for the minimum: the cosine has a minimum at the points π + 2kπ, where k is an integer and its value is −1.
• cosine is an even and bounded function.

Tangent

The tangent of an angle alpha is equal to the ratio of the length of the opposite branch to the length of the adjacent branch in a right triangle. We usually denote the tangent by either tg or tan.

$$\tan(\alpha)=\frac{\mbox{ The length of the opposite hanger }}{\mbox{ The length of the adjacent hanger }}$$

The graph of this function is a curve called the tangent:

The tangent differs from the previous functions mainly in the period, which is no longer 2π, but only one π. Other properties:

• Thedefining domain is the set of

$$D(tan)=\mathbb{R}-\left\{\frac{\pi}{2}+k\pi\right\}\quad k\in\mathbb{Z}$$

• The domain of values is the set of all real numbers.
• Tangent has no maximum or minimum and is not bounded.
• The tangent is an odd function.

Cotangent

The cotangent of an alpha angle is equal to the ratio of the length of the adjacent branch to the length of the opposite branch. We usually denote the cotangent by cot or cotan.

$$\cot(\alpha)=\frac{\mbox{ The length of the adjacent hanger }}{\mbox{ The length of the opposite hanger }}$$

The graph of this function is a curve called a cotangent:

Other properties:

• The period is also π, as for tangents.
• Thedefining domain is the set of

$$D(cot)=\mathbb{R}-\left\{k\pi\right\};\quad k\in\mathbb{Z}$$

• The domain of values is the set of all real numbers.
• The cotangent has no maximum or minimum and is not bounded.
• Cotangent is an odd function.

The relationship between sine and cosine

Goniometric functions have close relationships between them. If you look at the graph of the sine and cosine functions at the same time, you will see that they are not very different from each other, that one is just slightly offset from the other.

So if we always add π/2 to the argument of the function, we get the cosine function:

Conversely, if we subtract π/2 from the cosine , we get the sine.

So we can write these formulas:

$$\begin{eqnarray} \sin(x)&=&\cos(x-\frac{\pi}{2})\\ \cos(x)&=&\sin(x+\frac{\pi}{2}) \end{eqnarray}$$

How to express tangent and cotangent

We can easily express the tangent function in terms of the sine and cosine. Just remember what tangent is: the ratio of the length of the opposite branch to the length of the adjacent branch. Let's take this triangle to help us:

What does the tangent of the angle beta equal?

$$\tan(\beta)=\frac{|b|}{|c|}.$$

How could we express the numerator or denominator? What is the length of the side b and the length of the side c equal to ? We know that the sine of the angle beta is equal to:

$$\sin(\beta)=\frac{|b|}{|a|}$$

From that formula, we isolate |b|:

$$|b|=\sin(\beta)\cdot|a|$$

The cosine of the angle beta is equal to:

$$\cos(\beta)=\frac{|c|}{|a|}$$

We isolate |c|:

$$|c|=\cos(\beta)\cdot|a|$$

At this point, we have the side lengths b and c expressed using the sine cosine functions. So we plug these intermediate results into the original equation with the tangent:

$$\tan(\beta)=\frac{|b|}{|c|}=\frac{\sin(\beta)\cdot|a|}{\cos(\beta)\cdot|a|}$$

We can truncate |a| from the numerator and denominator to get the final formula:

$$\tan(\beta)=\frac{\sin(\beta)}{\cos(\beta)}$$

We can thus break down the tangent as the quotient of the sine and cosine. Already without derivation, we can write the cotangent as an inverse fraction, i.e., the quotient of cosine divided by sine.

$$\begin{eqnarray} \tan(\alpha)&=&\frac{\sin(\alpha)}{\cos(\alpha)}\\ \cot(\alpha)&=&\frac{\cos(\alpha)}{\sin(\alpha)} \end{eqnarray}$$

The relationship between tangent and cotangent

Just as the sine and cosine functions are similar, so are the tangent and cotangent functions. Look at both graphs in one picture:

How do we make the curve that describes the tangent a curve that describes the cotangent? The cotangent is shifted by π/2, so we start with that:

We've gotten closer to the cotangent, but we still have curves in the wrong direction, the cotangent is decreasing at selected intervals, while this shifted curve is increasing. We can help this by changing the sign of the variable x:

We can write:

$$\cot(x)=\tan(-x+\frac{\pi}{2})$$

Tabular values

Sine, cosine, tangent and cotangent have nice resulting values for some nice angles. Here is a basic overview of them:

$$ \LARGE \begin{matrix} &\sin&\cos&\tan&\cot\\ 0^\circ&0&1&0&\times\\ 30^\circ&\frac12&\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{3}&\sqrt{3}\\ 45^\circ&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&1&1\\ 60^\circ&\frac{\sqrt{3}}{2}&\frac12&\sqrt{3}&\frac{\sqrt{3}}{3}\\ 90^\circ&1&0&\times&0 \end{matrix} $$