Trigonometry Made Simple

Introduction to Trigonometry:
Trigonometry is a derived from a Greed word ‘tri’ (meaning three) and ‘gon’ (meaning sides and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on Trigonometry was recorded in Egypt and Babylon. Early Astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on Trigonometric concepts.

The Trigonometric Ratios of the angle A in right triangle ABC are defined as:
Sine of angle A = (side opposite to angle A)/hypotenuse; cosec = 1/Sine
Cosine of angle A = (side adjacent to angle A)/hypotenuse; sec = 1/cosine
Tangent of angle A = (side opposite to angle A)/(side opposite to angle A); cot =1/tan
The trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the lengths of its sides

Trigonometry Problems and Answers:
Problem: Given angle A=51 degrees, Adjacent side length = x and opposite side length = 10. Find x and H, hypotenuse of the triangle
Answer: tan(A) = opposite side/adjacent side
tan(51)= 10/x
X = 10/(tan 51) = 8.1 (two significant digits)
Sin(A) = opposite side/hypotenuse
Sin(51) = 10/H
H = 10/ Sin(51) = 13 (two significant digits)
x=8.1 and H=13

Problem: If sin 3A = cos (A- 26 degrees), where 3A is an acute angle, find the value of A
Answer:  sin 3A can be written as cos (90-3A)
      So, we get            cos (90-3A) = cos(A-26degrees)
Since both 90-3A and A-26 are both acute angles,
90 – 3A = A- 26
4A = 116
A = 29 degrees

Problem:  An observer 1.5m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from his eyes is 45 degrees. Calculate the height of the chimney.
Answer: Let us draw a rough triangle ABC with the right angle at B. let us draw a line DE parallel to BC such that AB (AE+EB) will be the height of the chimney,  CD (equal to BE) the observer and Angle ADE, the angle of elevation.  Here, ADE is the right triangle, right-angled at E
We have, AB = AE +EB = AE +1.5 DE = CB = 28.5 (distance from the chimney)
Let us use the tangent of the angle of elevation
tan(45degrees) = AE/DE
1 = AE/28.5
AE= 28.5
Height of the chimney = AE +EB =28.5 +1.5 = 30 m