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Trigonometry

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 Sub Topics Trigonometry is a branch of mathematics which deals with the real life problems related to angles. 'Trigo' means three sides. So, trigonometry is basically a branch of mathematics which relates to the study of three sided figures i.e. Triangles. Trigonometry is a branch of Math studying the relationship between angles and lengths of the sides. In trigonometry, we take into consideration, a Right Angle triangle with acute angles. These Relations are used to solve real life problems related to inclination at a particular point.Trigonometry is used to find the angles or the distance between the two points, when the angle of inclination is known. Trigonometric functions such as sine, cosine and tangent are used in computations in Trigonometry.Trigonometry is an important branch of mathematics which deals with ratios and relationships between angles and sides of triangles, especially right angled triangles. The study of trigonometry mainly revolves around the three functions. The values of sine, cosine and tangent vary with the change of angles. Example, sin (0) = 0, sin (30) = 1/2.

Trigonometry Definition

Trigonometry can be defined as the branch of mathematics, which deals with the relationship between the sides and angles of triangles based on the calculations, particularly using the trigonometric functions. This concept is concerned with the properties of the trigonometry functions and the application to determine the angles and sides of a triangle.

Trigonometric Functions

The Trigonometry functions are the functions which describes the function of angle. These angles are related to the triangles to the lengths of its sides. They are also called as circular functions in addition they make the core of trigonometry. With trigonometry we can find the height of a building or the width of a river without actually climbing or crossing it.
Trigonometric functions can also be expressed as the ratio of two of the sides of a right triangle that contains the angle, the sine, cosine, tangent, and co-tangent, secant and co-secant.
Sine, cosine, tangent are three functions that are simply ratios of the sides of triangles that help us relate to an angle in the triangle.
There are six different trigonometry functions, for a right triangle BAC, and 0$^o$ < $theta$ < 90$^o$

Sin $\theta$ = $\frac{\text{Perpendicular}}{\text{Hypotenuse}}$

Cosine $\theta$ = $\frac{\text{Base}}{\text{Hypotenuse}}$

Tangent $\theta$ = $\frac{\text{Perpendicular}}{\text{Base}}$

These three functions are the prime functions of trigonometry functions.

Cosec $\theta$ = $\frac{\text{Hypotenuse}}{\text{Perpendicular}}$

Secant $\theta$ = $\frac{\text{Hypotenuse}}{\text{Base}}$

Cotangent $\theta$ = $\frac{\text{Base}}{\text{Perpendicular}}$
These are the inverse functions of the first three prime trigonometry functions.

Trigonometry Identities

The equalities that involve trigonometric functions and true for every single value of the occurring variables are called as trigonometric identities. This identities involves one or more angles.

Here are few of the Trigonometric Identities:

Trigonometric Pythagorean Identities:
sin2 a + cos2 a = 1

1 + tan2 a = sec2 a

1 + cot2 a = csc2 a

Trigonometric Identities for Reciprocal:

csc a = $\frac{1}{\sin a}$

sec a = $\frac{1}{\cos a}$

cot a = $\frac{1}{\tan a}$

sin a = $\frac{1}{\csc a}$

cos a = $\frac{1}{\sec a}$

taa = $\frac{1}{\cot a}$

Trigonometric Identities for Quotient:

tan a = $\frac{\sin a}{\cos a}$

cot a = $\frac{\cos a}{\sin a}$

Trigonometric Identities for Even - Odd:

cos (-a) = cos a

sin (-a) = - sin a

tan (-a) = - tan a

sec (-a) = sec a

csc (-a) = - csc a

cot (-a) = - cot a

Trigonometric Identities for double Angle Formula:

sin (2a) = 2 sin a cos a

cos (2a) = 1 - 2 sin2 a

tan (2a) = $\frac{2 \tan a}{1 - \tan ^{2} a}$

Now, lets try proving few trigonometric identities.

We know that,

sin $\theta$ = $\frac{\text{opposite}}{\text{hypotenuse}}$

cos $\theta$ = $\frac{\text{adjacent}}{\text{hypotenuse}}$

tan $\theta$ = $\frac{\text{opposite}}{\text{adjacent}}$

According to Pythagorean theorem

a2 + b2 = c2

By dividing c2 both the side, we get

$\frac{a^{2}}{c^{2}}$ + $\frac{b^{2}}{c^{2}}$ = $\frac{c^{2}}{c^{2}}$

i.e. ${(\frac{a}{c})^2}$ + ${(\frac{a}{c})^2}$ = 1

cos2 $\theta$ + sin2 $\theta$ = 1.
Now, divide both the sides by cos2$\theta$

$\frac{\sin ^{2} \theta}{\cos ^{2} \theta}$ + $\frac{\cos ^{2} \theta}{\cos ^{2} \theta}$ = $\frac{1}{\cos ^{2} \theta}$

tan2 $\theta$ + 1 = sec2 $\theta$Hence proved.

Given below are few problems based on Trigonometry Identities:

Solved Examples

Question 1: Prove (cos A - sin A +1) / (Cos A + sin A - 1) = cosec A + cot A
Solution:
Let us divide the numerator and denominator of the left hand side by sin A

$\frac{\cos A - \sin A + 1}{\sin A}$

$\frac{\cos A + \sin A - 1}{\sin A}$

We get $\frac{\cot A - 1 + \csc A}{\cot A + 1 - \csc}$

= $\frac{\cot A - (1 - \csc A)}{\cot A + (1 - \csc A)}$

Now, let us multiply the numerator and denominator by cot A - (1 - cosec A)

$\frac{(\cot A - (1 - \csc A))^2}{\cot^2 A - (1 - \csc A)^2}$

=  $\frac{\cot^2 A + 1 + \csc^2 A - 2\csc A - 2\cot A + 2\cot A \csc A}{2\csc A - 2}$

$\frac{2 \csc\ A(\csc A - 1) + 2 \cot A(\csc A - 1)}{2 \csc A - 2}$

= $\frac{2(\csc A + \cot A)(\csc A - 1)}{2(\csc A - 1)}$

By cancelling the common terms, we get

(cos A - sin A + 1)/ (Cos A + sin A - 1) = cosec A + cot A.

Hence proved.

Question 2: Solve (sec A + tan A)(1 - sin A)
Solution:
(sec A + tan A)(1 - sin A) = sec A + tan A - sin A sec A - sin A tan A

= $(\frac{1}{\cos A})$ + $(\frac{\sin A}{\cos A})$ - sin A$(\frac{1}{\cos A})$ - sin A$(\frac{\sin A}{\cos A})$

= $\frac{1 - \sin^2 A}{\cos A}$

= $\frac{\cos^2 A}{\cos A}$

= cos A
Hence proved.