Basics of Trigonometry

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The study of angles and of the angular relationships of planar and three-dimensional figures is known as trigonometry. Trigonometry is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). It specifically deals with the relationships between the sides and the angles of triangles; the trigonometric functions, and calculations based on them.

Trigonometry is a methodology for finding some unknown elements of a triangle (or other geometric shapes) provided the data includes a sufficient amount of linear and angular measurements to define a shape uniquely. For example, two sides of a triangle and the angle they include define the triangle uniquely. The third side can then be found from the Law of Cosines while the angles are determined from the Law of Sines.

Trigonometric ratios

Trigonometry uses the fact that ratios of pairs of sides of triangles are functions of the angles. The basis for mensuration of triangles is the right-angled triangle. The term trigonometry means literally the measurement of trigons (triangles). This mensuration approach defines the six trigonometric ratios in terms of ratios of lengths of sides of a right triangle.

(a) sin θ = opp/hyp = AB/AC
(b) cos θ = adj/hyp =BC/AC
(c) tan θ = opp/adj = AB/BC
(d) cosec θ = hyp/opp = AC/AB
(e) sec θ = hyp/adj = AC/BC
(f) cot θ = adj/opp =BC/AB

From the ratios, we can easily observe the following relations.
(a) sin θ = 1/cosec θ
(b) cos θ = 1/sec θ
(c) tan θ = 1/cot θ
(d) tan θ = sin θ /cos θ
(e) cosec θ = 1/sin θ
(f) sec θ = 1/cos θ
(g) cot θ = 1/tan θ
(h) cot θ = cos θ/sin θ

Other Standard Results

(a) sin2θ + cos2θ = 1
(b) cosec2θ - cot2θ = 1
(c) sin 2θ = 2sinθcosθ
(d) cos 2θ = 2cos2θ  - 1 = 1 - 2sin2θ = cos2θ - sin2θ

In a triangle, if A, B, C are the angles opposite to the sides of the triangle a, b, c respectively and if R is the circumradius of the triangle, then
(e) a / sin A = b / sin B = c/sin C = 2R
(f) cos A = (b2 + c2 - a2) / 2bc
(g) cos B = (a + c2 - b2) / 2ac 
(h) cos B = (a + b2 - c2) / 2ab
(i) sin (A + B) = sin A cos B + cos A sin B
(j) sin (A – B) = sin A cos B – cos A sin B
(k) cos (A + B) = cos A cos B – sin A sin B
(l) cos (A – B) = cos A cos B + sin A sin B
(m) tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
(n) tan (A – B) = (tan A – tan B) / (1 + tan A tan B)
(o) sin 2A = 2 sin A cos A = 2 tan A / (1 + tan 2A)
(p) cos 2A = cos2A – sin2A = (1 – tan2A) / (1 + tan2A)=1 – 2 sin2A = 2 cos2A – 1

Solving Right Angled Triangles
One of the most important applications of trigonometric functions is to solve a right triangle. Trigonometric functions can be used to solve a right triangle if either the length of one side and the measure of one acute angle is known or the length of two sides is known. The angle between the line of sight and horizontal is called the angle of elevation if the object of observation is at a higher level than the eye.

The angle between the line of sight and horizontal is called the angle of depression if the object of observation is at a lower level than the eye.

Solving non-right angled triangles
Trigonometry can also be used to solve a non-right angled triangle. Laws of sines and laws of cosines (Other standard formulas – points v to viii) can be used to solve non-right angled triangles provided one of the following information is known:
a. two sides and an angle opposite one of the known sides
b. two angles and any side
c. two sides and their included angle
d. all three sides