Three Functions, Same Idea.

Right-Angled Triangle:

"Opposite" is opposite to the angle θ
"Adjacent" is adjacent (next to) to the angle θ
"Hypotenuse" is the longest side

Before we start with the three trigonometric functions - sine, cosine and tangent, we will first look at a summary of them below:

These circular functions are functions of an angle, used to relate the angles of a triangle to the lengths of the sides of a triangle. The functions are important in the study of triangles and modeling periodic phenomena, among other applications.

The most familiar trigonometric functions are the sine, cosine, and tangent. Below is a description for each function:

Sine: taking an angle and finding the length of the y-component (rise) of that particular right-angled triangle.

Cosine: taking an angle and finding the length of x-component (run) of that particular right-angled triangle.

Tangent: taking an angle and finding the slope (y-component divided by the x-component) of that particular right-angled triangle.

Trigonometric functions are usually represented in ratios of two sides of a particular right-angled triangle, like the following:

Using the functions is relatively simple - you only need to know either the two other sides or one side with the theta.

Thus, for a right-angle triangle with an angle θ , the functions can be calculated:

Sine Function: sin(θ) = Opposite / Hypotenuse
Cosine Function: cos(θ) = Adjacent / Hypotenuse
Tangent Function: tan(θ) = Opposite / Adjacent

Here are 2 examples:

Example 1: what are the sine, cosine and tangent of 30° ?

The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of √(3):
Now we know the lengths, we can calculate the functions:Sine sin(30°) = 1 / 2 = 0.5
Cosine cos(30°) = 1.732 / 2 = 0.866
Tangent tan(30°) = 1 / 1.732 = 0.577

Example 2: what are the sine, cosine and tangent of 45° ?

The classic 45° triangle has two sides of 1 and a hypotenuse of √(2):

Sine sin(45°) = 1 / 1.414 = 0.707
Cosine cos(45°) = 1 / 1.414 = 0.707
Tangent tan(45°) = 1 / 1 = 1

Please refer to the following GSP file for evidence/proof:

Download Trigonometry Functions GSP

References:

1. Wikipedia

2. Wikispaces

3. Maths Is Fun

4. Interactive Mathematics

Trigonometry Functions

Many aspects of our lives use Mathematics and Trigonometry. Be it engineering, information communication or surveying, trigonometry has certainly helped us in our jobs and in solving daily problems.

The basic building block of trigonometry is angle. It is defined as a figure formed by two rays sharing a vertex, thus it is also the measurement of the amount of rotation between two line segments.

The two rays drawn from the vertex are named the terminal side and the initial side. Angles are measured in either degrees (°) or radians. 1 radian is about 57.3°. Radians are much more useful and convenient in science and engineering, as well as in calculus.

Interactive Mathematics defines the standard position of an angle as:

"An angle is in standard position if the initial side is the positive x-axis and the vertex is at the origin. The 2 examples given above are in standard position.

We will use r, the length of the hypotenuse, and the lengths x and y when defining the trigonometric ratios..."