Trigonometry Definitions

Right-angled triangle definitions

In this way of looking at things, basic trigonometric functions are defined as ratios of the lengths of the sides of a right triangle.

Figure 1

Figure 1 shows a right triangle i.e., a triangle where one of its angles = 90º. Such an angle is referred to as a right angle, thus the name. The triangle above consists of 2 sides, a and b, that join to form the right angle. The little box nestled in the angle formed by sides a and b indicates that angle is a right angle. The side opposite the right angle, c, is called the hypotenuse of the triangle. Sides a and c form the angle \theta. Sides b and c form angle \phi. Side a is said to be adjacent to \angle\theta. Side b is said to be adjacent to \angle \phi.

Given this setup, the following functions can be defined for \angle \theta:

Sine (= sin)

(1)   \begin{equation*}\sin \theta = \frac {\text{side opposite to }\, \theta}{\text{hypotenuse}}=\frac bc=\frac{1}{\csc}\end{equation*}

Cosine (= cos)

(2)   \begin{equation*}\cos \theta = \frac {\text{side adjacent to }\, \theta}{\text{hypotenuse}}=\frac ac=\frac{1}{\sec}\end{equation*}

Tangent (= tan)

(3)   \begin{equation*}\tan \theta = \frac {\text{side opposite to }\, \theta}{\text{side adjacent to }\,\theta}=\frac ba=\frac{\sin\theta}{\cos\theta}\end{equation*}

Cotangent (= cot)

(4)   \begin{equation*}\cot \theta = \frac {\text{side adjacent to }\, \theta}{\text{side opposite to }\,\theta}=\frac ab=\frac{\cos\theta}{\sin\theta}\end{equation*}

Secant (= sec)

(5)   \begin{equation*}\sec \theta = \frac {\text{hypotenuse}}{\text{side adjacent to }\, \theta}=\frac ca=\frac{1}{\cos\theta}\end{equation*}

Cosecant (= csc)

(6)   \begin{equation*}\csc \theta = \frac {\text{hypotenuse}}{\text{side opposite to }\, \theta}=\frac cb=\frac{1}{\sin\theta}\end{equation*}

Figure 2

Unit-circle definitions

In this way of looking at things, the trigonometric functions are defined as coordinate values of points in the Euclidean plane that are related to a unit circle. A unit circle, in turn, is a circle with its center at the origin of a Euclidean coordinate system and a radius of length equal 1 unit. The radius of a circle, of course, is the length of a line segment connecting the center of the circle and any point on the circle. In figure 2, AC is a radius. It is also the hypotenuse of triangle ABC.

Also,

  • The x-coordinate of point C is AB=x
  • \cos \theta = \frac {AB}{AC}
  • AC=1\,\therefore\,\cos \theta = \frac {AB}{1}=\frac {x}{1}=x, the x-coordinate of C
  • The y-coordinate of point C is BC.
  • \sin \theta = \frac {BC}{AC}
  • AC=1\,\therefore\,\sin \theta = \frac {BC}{1}=\frac {y}{1}=y, the y-coordinate of C
  • Since x=\cos\theta and y=\sin\theta, the coordinates of the point C are (\cos\theta,\,\sin\theta)

An excellent summary of this topic can be found at Khan Academy.

Graphs of Trigonometric Function

Graphs of the basic trigonometric functions can be found at SparkNotes.