Introduction to Trigonometry
This chapter introduces the trigonometric ratios of acute angles in a right triangle, their values for standard angles (0, 30, 45, 60, 90 degrees), and the fundamental trigonometric identities.
In a right triangle with an acute angle theta: sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, tan(theta) = opposite/adjacent. The reciprocals give cosec(theta), sec(theta) and cot(theta) respectively.
sin: 0 degrees = 0, 30 = 1/2, 45 = 1/root2, 60 = root3/2, 90 = 1. cos: 0 = 1, 30 = root3/2, 45 = 1/root2, 60 = 1/2, 90 = 0. tan: 0 = 0, 30 = 1/root3, 45 = 1, 60 = root3, 90 = undefined.
Q. In a right triangle ABC, right-angled at B, if AB = 3 cm and BC = 4 cm, find sin A and cos A.
By Pythagoras theorem, AC = root(AB^2 + BC^2) = root(9 + 16) = root(25) = 5 cm. sin A = (side opposite to A)/hypotenuse = BC/AC = 4/5. cos A = (side adjacent to A)/hypotenuse = AB/AC = 3/5.
Fundamental trigonometric identities: sin^2(theta) + cos^2(theta) = 1; 1 + tan^2(theta) = sec^2(theta); 1 + cot^2(theta) = cosec^2(theta). These hold for all values of theta where the functions are defined.
Q. Evaluate: 2 tan^2(45) + cos^2(30) - sin^2(60).
tan(45) = 1, so tan^2(45) = 1, and 2 tan^2(45) = 2. cos(30) = root3/2, so cos^2(30) = 3/4. sin(60) = root3/2, so sin^2(60) = 3/4. Total = 2 + 3/4 - 3/4 = 2.
The value of tan(90 degrees) is:
Correct answer: (D) Undefined (not defined)
sin(theta) x cosec(theta) = _______.
1
The value of sin(theta) increases as theta increases from 0 to 90 degrees.
True — sin(theta) increases monotonically from 0 (at 0 degrees) to 1 (at 90 degrees) as theta increases through this range.