 ## TRIGONOMETRY

Trigonometry

It is the study of triangles where we deal with the angles and sides of the triangle. To be more specific, its 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 by the help of trigonometric ratios. The angles are either measured in radians or degrees. The usual trigonometry angles are 0°, 30°, 45°, 60° and 90°, which are commonly used.

Trigonometry Ratios-Sine, Cosine, Tangent

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. 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.

Trigonometric Functions:

 Function Abbreviation Relationship to sides of a right triangle Sine Function sin Opposite side/ Hypotenuse Tangent Function tan Opposite side / Adjacent side Cosine Function cos Adjacent side / Hypotenuse Cosecant Function cosec Hypotenuse / Opposite side Secant Function sec Hypotenuse / Adjacent side Contangent Function cot Adjacent side / Opposite side

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

Formulae involving multiple and submultiples angles

Formulae involving multiple angles

• sinx*cosy = sin(x+y) + sin(x−y)2
• cosx*cosy = cos(x+y) + cos(x−y)2
• sinx*siny = cos(x+y) − cos(x−y)2

Double Angle Formulas

• sin2θ = 2 sinθ cosθ
• cos2θ = cos2θ−sin2θ
• tan2θ = (2tanθ)/(1−tan2θ)
• cot2θ = (cot2θ−1)/(2cotθ)

Triple Angle Formulas

• sin3θ = 3sinθ − 4sin3θ
• cos3θ = 4cos3θ − 3cosθ
• tan3θ = (3tanθ−tan3θ)/(1−3tan2θ)
• cot3θ = (3cotθ−cot3θ)/(1−3cot2θ)

Formulae submultiples angles

• sin A = 2 sin A/2 cos A/2
• cos A = cos2 A/2A – sin2 A/2
• cos A = 2 cos2 A/2 - 1
• cos A = 1 - 2 sin2 A/2
• 1 + cos A = 2 cos2 A/2
• 1 - cos A = 2 sin2 A/2
• tan2 A/2 = (1−cosA)/(1+cosA)

General solution of trigonometric equations:

1. Find the general solution of the equation cos2x-2tan x+2=0
Note that cos 2x = (1-tan2 x)/(1+tan2x)
Substituting this, the equation has only one variable: tan x. We evaluate and then factorize the expression to obtain
(tan x - 1)(2tan2 x + tan x + 3) = 0
As the equation 2tan2 x + tan x + 3 = 0 has imaginary roots, we do not consider it. Hence, we have
tan x - 1 = 0
tan x = 1
x = π/4 + n π

2. What are the solutions of the equation Sin2x = √3/2

In the x – interval [0,2 π]
Since 0 ≤ 2x ≤ 4 π, we have
2x = π/3, 2/3 π, 7/3 π, 8/3 π
X = π/6, π/3, 7/6 π, 4/3 π
3. What is the general solution of tan x = 1
In the first period 0 ≤ x < π the solution for this equation is x = π /4. Since the period of tan x is π, the general
solution of the given equation is
X = n π + n/4

## NATA Exam MCQ for PHYSICS

Properties of triangles

The sum of all the angles of a triangle(of all types) is equal to 1800.
The sum of the length of the two sides of a triangle is greater than the length of the third side.
In the same way, the difference between the two sides of a triangle is less than the length of the third side.
The side opposite the greater angle is the longest side of all the three sides of a triangle.
The exterior angle of a triangle is always equal to the sum of the interior opposite angles. This property of a
triangle is called an exterior angle property
Two triangles are said to be similar if their corresponding angles of both triangles are congruent and lengths
of their sides are proportional.

Area of a triangle = ½ × Base × Height
The perimeter of a triangle = sum of all its three sides

Types of triangles

Scalene Triangle: All the sides and angles are unequal.
Isosceles Triangle: It has two equal sides. Also, the angles opposite these equal sides are equal.
Equilateral Triangle: All the sides are equal and all the three angles equal to 600.
Acute Angled Triangle: A triangle having all its angles less than 900.
Right Angled Triangle: A triangle having one of the three angles is 900.
Obtuse Angled Triangle: A triangle having one of the three angles more than 900.

Inverse trigonometric functions and their properties

Inverse trigonometric functions are also called “Arc Functions” since, for a given value of
trigonometric functions, they produce the length of arc needed to obtain that particular value. The inverse trigonometric functions actually perform the opposite operation of the trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. We know that, trig functions are specially applicable to the right angle triangle. These six important functions are used to find the angle measure in a right triangle when two sides of the triangle measures are known

Properties

Property Set 1:

Sin−1(x) = cosec−1(1/x), x∈ [−1,1]−{0}
Cos−1(x) = sec−1(1/x), x ∈ [−1,1]−{0}
= cot−1(1/x) −π, if x < 0
Cot−1(x) = tan−1(1/x), if x > 0 Or,
= tan−1(1/x) + π, if x < 0

Property Set 2:

Sin−1(−x) = −Sin−1(x)
Tan−1(−x) = −Tan−1(x)
Cos−1(−x) = π − Cos−1(x)
Cosec−1(−x) = − Cosec−1(x)
Sec−1(−x) = π − Sec−1(x)
Cot−1(−x) = π − Cot−1(x)

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