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The branch of mathematics dealing with the relationships between the angles and the sides of triangles, and the calculations based on them. It originated as the study of certain mathematical relations in a triangle containing a right angle (90°). Literally, it means triangle measurement, though its applications extend far beyond geometry. It emerged as a rigorous discipline in the 15th century, when the demand for accurate surveying techniques and navigational methods led to its use for the “solution” of right triangles, or the calculation of the lengths of two sides of a right triangle given one of its acute angles and the length of one side. The solution can be found by using ratios in the form of the trigonometric functions.

There are an enormous number of applications of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.

FUNDAMENTALS OF TRIGONOMETRY

MEASURING ANGLES

Two of the units that we normally use to measure angles in a plane are degrees and radians. One radian is the angle suspended at the centre of a circle by an arc of length r, where r is the radius of the circle. A circle consists of 360^O or 2p rad.

1 radian = 180^O/p

Therefore 1^O = p/180 radians

Trigonometric ratios are defined in a right angle triangle system only. The three trigonometric ratios are defined as (refer figure)

For the angle pictured in the figure, we see that

 

0	0	0	1	0
				1
		1	0	(undefined)

TRIGONOMETRIC RATIOS

1. The ratio of the perpendicular to the hypotenuse is called the sine of the angle

q

Thus, sin q = AC/AB

2. The ratio of the base to the hypotenuse is called the cosine of the angle q.

cos q = BC/AB

3. The ratio of the perpendicular to the base is called the tan of the angle

q.

Thus, tan q = AC/BC.

The reciprocal ratios are defined as

cosec q = 1/sinq

secq = 1/cosq,

cotq = 1/tanq

SQUARE RELATIONS

• sin²

q + cos²q = 1

• sec²

q = 1 + tan²q

• cosec²

q = 1 + cot²q

FUNCTIONS OF ANGLES OF ANY MAGNITUDE

• sin (-q) = -sinq
• cos (-q) = cosq
• sin (p/2 - q) = cosq
• cos (p/2 - q) = sinq
• sin (p/2 + q) = cosq
• cos (p/2 + q) = - sinq
• sin (p - q) = sinq
• cos (p - q) = - cosq
• sin (p + q) = - sinq
• cos (p + q) = - cosq