Trigonometry

Right Angled Triangles |  Trigonometric Ratios |  Special Angles  |  Inverse Trigonometric Functions  |  Finding the Length of a Side of a Triangle  |  Trigonometric Identities  |  Miscellaneous  

Right Angled Triangles

If one of the angles of a triangle is 90º (a right angle), the triangle is called a right angled triangle. We indicate the 90º (right) angle by placing a box in its corner. (See diagram below.) Because the three (internal) angles of a triangle add up to 180º, the other two angles are each less than 90º; that is they are acute.

In the above triangle, the side H opposite the right angle is called the hypotenuse. Relative to the angle θ, the side O opposite the angle θ is called the opposite side. The remaining side A is called the adjacent side.

Warning: This assignment of the opposite and adjacent sides is relative to θ. If the angle of interest (in this case θ) is located in the upper right hand corner of the above triangle the assignment of sides is then as shown below.

Pythagoras Theorem states that a triangle is right angled if and only if

.

This means that given any two sides of a right angled triangle, the third side is completely determined.

For example, if O = 1, A = 2, then .

If H = 5, and O = 3, then .

Trigonometric Ratios

Trigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are

sine cosine tangent

Note that .

Other ratios are defined by using the above three:

These six ratios define what are known as the trigonometric (trig in short) functions. They are independent of the unit used.

Exercise 1

Given the lengths of the three sides of a right angled triangle find the values of the trig functions, corresponding to the angle θ. (Round your answers to 2 decimal places)     H = O =   A =   sin θ = 2 d.p. cos θ = 2 d.p. tan θ = 2 d.p. csc θ = 2 d.p. sec θ = 2 d.p. cot θ = 2 d.p.

Exercise 2.

Given the lengths of two sides of a right angled triangle find the length of the third side (use Pythagoras Theorem). Then find the values of the given trig functions corresponding to the angle θ. (Round your answer to 2 decimal places)     H = O =   A =   sin θ = 2 d.p. cos θ = 2 d.p. tan θ = 2 d.p.

Next - Special Angles

Right Angled Triangles |  Trigonometric Ratios |  Special Angles  |  Inverse Trigonometric Functions  |  Finding the Length of a Side of a Triangle  |  Trigonometric Identities  |  Miscellaneous