4.2 Trigonometric Ratios and Special Angles

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.






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

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.

Trigonometric Ratios of Special Angles

We have already learnt that trigonometry is the study of relationships between the sides and angles of a triangle. The ratios of the sides in a right triangle with respect to its acute angles are called trigonometric ratios of the angle. The angles of degree 0, 30, 45, 60 and 90  are useful angles in trigonometry, and their numerical values are easy to remember. When two angles add up to 900, then any one angle is the complement of the other. Trigonometric ratios of complementary angles help in simplifying problems. Hopefully this figures will help you guys. Remember to memorize :))