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Trigonometry

Trigonometric Ratios

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.) 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:

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 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.

These ratios are independent of the unit used to measure the sides as long as the same unit is used for all the sides.

In particular, if we take H = 1, then

O = sin θ  and  A = cos θ.

Special Angles

The trigonometric ratios of the angles 30º, 45º and 60º are often used in mechanics and other branches of mathematics. So it is useful to calculate them and know their values by heart.

45º

In this case, the triangle is isosceles. Hence the opposite side and adjacent sides are equal, say 1 unit.
The hypotenuse is therefore of length units (by Pythagoras Theorem).

We have

60º & 30º 

Let us draw an equilateral triangle, ABC, of sides 2 units in length. Next draw a line AD from A perpendicular to BC.  AD bisects BC giving BD = CD = 1.

From this we can determine the following trig ratios for the special angles 30º and 60º:

General Angles

For any other angle θ, you can calculate approximately the values of sin θ, cos θ, tan θ by using a scientific calculator.

Make sure you set the mode on your calculator to DEG if the angle is measured in degrees or RAD if the angle is measured in radians.

Exercise 3

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