Introduction to Trigonometry
Trigonometry is one of the important branches of mathematics and this concept is given by a Greek mathematician Hipparchus. Basically, trigonometry is the study of triangles where we deal with the angles and sides of the triangle. To be more specific, trigonometry is all about a right-angled triangle. It is one of those divisions in mathematics that helps in finding the angles and missing sides of a triangle. In trigonometry, the angles are either measured in radians or degrees.
This branch divides into two sub-branches called plane trigonometry and spherical geometry. Trigonometry, in general, is about the trigonometric formulas, Â trigonometric functions, and ratios, Right-Angled Triangles, etc.
Trigonometric Functions and Ratios
The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos, and tan. Let us see how are these ratios or functions, evaluated in the case of a right-angled triangle.
Right Angled Triangle
Consider a right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse is referred to as the adjacent and opposite.
The trigonometric ratios are calculated by the below formulas using above figure.
Functions |
Abbreviation |
Relationship to sides of a right triangle |
Sine Function | sin | Opposite / Hypotenuse |
Tangent Function | tan | Opposite / Adjacent |
Cosine Function | cos | Adjacent / Hypotenuse |
Cosecant Function | cosec | Hypotenuse / Opposite |
Secant Function | sec | Hypotenuse / Adjacent |
Cotangent Function | cot | Adjacent / Opposite |
Trigonometric Ratios of Special Angles
Unit Circle
The concept of unit circle helps us to measure the angles of cos, sin and tan directly since the centre of the circle is located at the origin and radius is 1. Consider theta be an angle then,
Trigonometric Identities
The Trigonometric Identities are the equations which are true in the case of Right Angled Triangles. Some of the special trigonometric identities are as given below â€“
- Pythagorean Identities-
- sin Â² Î¸ + cos Â² Î¸ = 1
- tan ^{2} Î¸ + 1 = sec^{2} Î¸
- cot^{2} Î¸ + 1 = cosec^{2} Î¸
- sin 2Î¸ = 2 sin Î¸ cos Î¸
- cos 2Î¸ = cosÂ² Î¸ â€“ sinÂ² Î¸
- tan 2Î¸ = 2 tan Î¸ / (1 â€“ tanÂ² Î¸)
- cot 2Î¸ = (cotÂ² Î¸ â€“ 1) / 2 cot Î¸
- Sum and Difference identities-
For angles u and v, we have the following relationships:
sin(u + v) = sin(u)cos(v) + cos(u)sin(v)
cos(u + v) = cos(u)cos(v) â€“ sin(u)sin(v)
tan(u+v) = \(\frac{tan(u)\ +\ tan(v)}{1-tan(u)\ tan(v)}\)
sin(u â€“ v) = sin(u)cos(v) â€“ cos(u)sin(v)
cos(u â€“ v) = cos(u)cos(v) + sin(u)sin(v)
tan(u-v) = \(\frac{tan(u)\ -\ tan(v)}{1+tan(u)\ tan(v)}\)
- If A, B and C are angles and a, b and c are the sides of a triangle, then
Sine Laws
a/sinA = b/sinB = c/sinC
Cosine Laws
\(c^{2} = a^{2} + b^{2} – 2ab\ cos\ C\) \(a^{2} = b^{2} + c^{2} – 2bc\ cosA\ and\) \(b^{2} = a^{2} + c^{2} – 2ac\ cos\ B\)Applications of Trigonometry
Trigonometry finds its applications in various fields like oceanography, seismology, meteorology, physical sciences, astronomy, acoustics, navigation, electronics, etc.
It is also helpful to measure the height of the mountain, find the distance of long rivers, etc.
Trigonometry Word Problems
Example 1: Two friends, Rakesh and Vishal started climbing a pyramid shaped hill. Rakesh climbs 315 mtr and finds that the angle of depression is 72.3 degrees from his starting point. How high he is from the ground.
Solution: Let m is the height above the ground.
To find: Value of m
To solve m, use sine ratio.
Sin 72.3^{0} = m/315
0.953 = m/315
m= 315 x 0.953
m=300.195 mtr
The man is 300.195 mtr Â above the ground.
Example 2: Â A man is observing a pole of height 55 foot. According to his measurement, pole cast a 23 feet long shadow. Can you help him to know the angle of elevation of the sun from the tip of shadow?
Solution:
Let x be the angle of elevation of the sun, then
tan x Â = 55/23 = 2.391
x = tan^{-1}(2.391)
or x = 67.30 degrees
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