## Trigonometric Functions for a Right Angle

March 9, 2012 in Additional Mathematics, Right Angle, Trigonometry by Shivakkumar Vadiveyl

## Trigonometric Functions of a Right Angle

The trigonometric functions of a right angle states that

$begin{array}{ccccc}& sine,theta & = & dfrac{length, of, opposite, side}{length, of, hypotenuse} & vspace{2mm}& cosine,theta & = & dfrac{length, of, adjacent, side}{length, of, hypotenuse} & vspace{2mm}& tangent,theta & = & dfrac{length, of, opposite, side}{length, of, adjacent, side} & vspace{2mm}end{array}$

Trigonometry – Right Angles

Basic Trigonometric Functions

In the below diagram, we have a ray that has formed an angle $theta$. If we were to draw a (x,y) coordinate on the ray, we will get a right angled triangle.

The basic trigonometric functions states that

$begin{array}{ccccc}& sine,theta & = & dfrac{y}{r} & vspace{2mm}& cosine,theta & = & dfrac{x}{r} & vspace{2mm}& tangent,theta & = & dfrac{y}{x} & vspace{2mm}end{array}$

Trigonometry – Right Angles

r here is the radian of the ray and is also the hypotenuse of the right angle. It can be found using the pythagoras theorem,

$begin{array}{ccccc}& r & = & sqrt{x^{2}+y^{2}}end{array}$

Trigonometry – Right Angle Math Exercise

1) Find the sin, cos and tan of an angle formed by a ray originating from (0,0). It is given that the coordinate (4,3) passes through the ray.

First, let’s plot a simple graph with the given information.

Trigonometry – Right Angles Math Worksheet

The length of the radian, r,

$begin{array}{ccccc}& r & = & sqrt{x^{2}+y^{2}}& & = & sqrt{4^{2}+3^{2}}& & = & sqrt{16+9}& & = & sqrt{25}& r & = & 5end{array}$

Using the formula for sin, cos and tan as follows,

$begin{array}{ccccc}& sine,theta & = & dfrac{y}{r} & vspace{2mm}& cosine,theta & = & dfrac{x}{r} & vspace{2mm}& tangent,theta & = & dfrac{y}{x} & vspace{2mm}end{array}$

$begin{array}{ccccc}& sin,theta & = & dfrac{y}{r} & vspace{2mm}& & = & dfrac{3}{5}end{array}$

$begin{array}{ccccc}& cos,theta & = & dfrac{x}{r} & vspace{2mm}& & = & dfrac{4}{5}end{array}$

$begin{array}{ccccc}& tan,theta & = & dfrac{y}{x} & vspace{2mm}& & = & dfrac{3}{4}end{array}$