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Trigonometry

# Definition and Graphs of Trigonometric Functions

Angles (arguments of functions): $$\alpha$$, $$x$$
Trigonometric functions: $$\sin \alpha$$, $$\cos \alpha$$, $$\tan \alpha$$, $$\cot \alpha$$, $$\sec \alpha$$, $$\csc \alpha$$
Set of real numbers: $$\mathbb{R}$$
Coordinates of points on a circle: $$x$$, $$y$$

Radius of a circle: $$r$$
Integers: $$k$$

1. Trigonometric functions are elementary functions, the argument of which is an angle. Trigonometric functions describe the relation between the sides and angles of a right triangle. Applications of trigonometric functions are extremely diverse. For example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear in the solution of differential equations and functional equations.
2. The trigonometric functions include the following $$6$$ functions: sine, cosine, tangent, cotangent, secant, and cosecant. For each of these functions, there is an inverse trigonometric function.
3. The trigonometric functions can be defined using the unit circle. The figure below shows a circle of radius $$r = 1$$. There is a point $$M\left( {x,y} \right)$$ on the circle. The angle between the radius vector $$OM$$ and the positive direction of the $$x$$-axis is equal to $$\alpha$$.
1. The sine of an angle $$\alpha$$ is the ratio of the $$y$$-coordinate of the point $$M\left( {x,y} \right)$$ to the radius $$r$$:
$$\sin \alpha = y/r$$.
Since $$r = 1$$, the sine is equal to the $$y$$-coordinate of the point $$M\left( {x,y} \right)$$.
2. The cosine of an angle $$\alpha$$ is the ratio of the $$x$$-coordinate of the point $$M\left( {x,y} \right)$$ to the radius $$r$$:
$$\cos \alpha = x/r$$
3. The tangent of an angle $$\alpha$$ is the ratio of the $$y$$-coordinate of the point $$M\left( {x,y} \right)$$ to the $$x$$-coordinate:
$$\tan \alpha = y/x,\;$$ $$x \ne 0$$
4. The cotangent of an angle $$\alpha$$ is the ratio of the $$x$$-coordinate of the point $$M\left( {x,y} \right)$$ to the $$y$$-coordinate:
$$\cot \alpha = x/y,\;$$ $$y \ne 0$$
5. The secant of an angle $$\alpha$$ is the ratio of the radius $$r$$ to the $$x$$-coordinate of the point $$M\left( {x,y} \right)$$:
$$\sec \alpha = r/x = 1/x,\;$$ $$x \ne 0$$
6. The cosecant of an angle $$\alpha$$ is the ratio of the radius $$r$$ to the $$y$$-coordinate of the point $$M\left( {x,y} \right)$$:
$$\csc \alpha = r/y = 1/y,\;$$ $$y \ne 0$$
7. Relation between the sides and angles in a right triangle
In a unit circle, the projections $$x$$, $$y$$ of a point $$M\left( {x,y} \right)$$ and the radius $$r$$ form a right-angled triangle, in which $$x,y$$ are the legs, and $$r$$ is the hypotenuse. Therefore, the definitions given above are stated as follows:
The sine of an angle $$\alpha$$ is the ratio of the opposite leg to the hypotenuse.
The cosine of an angle $$\alpha$$ is the ratio of the adjacent leg to the hypotenuse.
The tangent of an angle $$\alpha$$ is the ratio of the opposite leg to the adjacent leg.
The cotangent of an angle $$\alpha$$ is the ratio of the adjacent leg to the opposite leg.
The secant of an angle $$\alpha$$ is the ratio of the hypotenuse to the adjacent leg.
The cosecant of an angle $$\alpha$$ is the ratio of the hypotenuse to the opposite leg.
8. Graph of the sine function
$$y = \sin x$$, domain: $$x \in \mathbb{R}$$, range: $$-1 \le \sin x \le 1$$
1. Graph of the cosine function
$$y = \cos x$$, domain: $$x \in \mathbb{R}$$, range: $$-1 \le \cos x \le 1$$
1. Graph of the tangent function
$$y = \tan x$$, domain: $$x \in \mathbb{R},$$ $$x \ne \left( {2k + 1} \right)\pi/2$$, range: $$- \infty \lt \tan x \lt \infty$$
1. Graph of the cotangent function
$$y = \cot x$$, domain: $$x \in \mathbb{R},$$ $$x \ne k\pi$$, range: $$- \infty \lt \cot x \lt \infty$$
1. Graph of the secant function
$$y = \sec x$$, domain: $$x \in \mathbb{R},$$ $$x \ne \left( {2k + 1} \right)\pi/2$$, range: $$\sec x \in$$ $$\left( {-\infty , -1} \right] \cup \left[ {1,\infty } \right)$$
1. Graph of the cosecant function
$$y = \csc x$$, domain: $$x \in \mathbb{R},$$ $$x \ne k\pi$$, range: $$\csc x \in$$ $$\left( {-\infty , -1} \right] \cup \left[ {1,\infty } \right)$$