Trigonometry - Mathematical Tools for Physics Class 11 (Concepts and Questions)

We are already familiar with the trigonometric functions we learned in previous classes — the trigonometric ratios and basic trigonometric identities. In this article, we shall go one step further and learn about the essential mathematical tools regarding trigonometric functions that we will apply for Class 11 physics for NEET 2023/2024.

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Table of Contents:

• Functions
• Algebraic Functions
  • Linear Functions
  • Quadratic Functions
• Trigonometric Functions
  • Trigonometric Ratios and Formulae

Functions

In simple words, a function is a rule that depicts some kind of relationship between two variables such that there is a unique output to each input variable.

Mathematically, if A and B are two non-empty sets such that ‘f’ is a relation that is defined between the given sets, it is said to be a function if, for every element in A, there exists a unique f-image in B.

We write, f: A → B such that f(a) = b ∈ B, ∀ a ∈ A.

Set A is called the domain of function f, and set B is called the co-domain of the function.

The set defined by

{b ∈ B | f(a) = b ∀ a ∈ A }

is called the range of f.

Let us understand how functions are applied in Physics. Consider a body of mass ‘m’, which is moving with a velocity ‘v’. Now, the kinetic energy of the body will depend on the velocity of the body as mass remains constant.

Then we can express the kinetic energy of a body as a function of velocity.

f(v) = ½ mv², where m is the constant mass of the body.

Thus a function could be considered as a machine with an input and a unique output. The inputs are the values from the domain of the function whereas outputs are the values from the range of the function.

We shall now discuss the various kinds of functions that we often use while dealing with problems in physics.

Algebraic Functions

In algebraic functions, the rule must be a polynomial of degree n. An algebraic function f(x) is defined as

f(x) = a_nxⁿ + a_{n – 1}x^{n – 1} + … + a₁x¹ + a_o, where a_i’s are real numbers, with a_n ≠ 0 called the coefficients.

An algebraic function can be of the form:

• Quadratic function
• Linear function
• Cubic function
• Bi-quadratic function
• Equation with more than one variable, etc.

Thus, any polynomial function is regarded as an algebraic function.

Linear functions

Algebraic functions with a polynomial of degree 1 are called linear functions. For example, the equation of a straight line is given by y = mx + c which is a linear polynomial. The graphical representation of this function will be a straight line with slope m and c intercept on the y-axis.

If the degree of the polynomial function is zero then it will represent a constant function. For example, x = a where ‘a’ is a constant. This will graphically represent a line that is parallel to the y-axis. Similarly, if we take y = b, where b is any constant, then this will represent a line parallel to the x-axis.

If we have a linear equation of the form y = x, this represents a straight line passing through the origin inclined equally to both axes.

Quadratic Functions

Algebraic functions which represent a polynomial of degree 2 are called quadratic functions. The general form of quadratic functions is ax² + bx + c, a ≠ 0. A quadratic equation ax² + bx + c = 0 has two roots.

Nature of roots depends on the value of discriminant D = b² – 4ac.

We can easily determine roots of a quadratic equation by factorisation or by using the quadratic formula which is also known Shridharacharya’s formula given by:

If 𝛼 and 𝛽 are the roots of the quadratic equation, then,

If S is the sum of the roots and P is the product of roots then the quadratic equation is written as,

x² – Sx + P = 0

A quadratic equation represents a parabola, and the points where the parabola cuts the x-axis are the roots of the quadratic equation.

Trigonometric Functions

Functions involving trigonometric ratios like sine, cosine, tangent, etc. are called trigonometric functions. For example, the wave function is given by y(x, t) = A sin(kx – ⍵t + ɸ) which is used to identify the position, acceleration and velocity of the particle of the medium in which the wave is traveling at any instant ‘t’.

Trigonometric functions are used to

• Measure the height and distance of objects.
• Represent the periodic nature of physical quantities.
• Do vector analysis of physical quantities.

Trigonometric Ratios

For a given right-angled triangle, the trigonometric ratios are given as:

Trigonometric values for common angles:

The three basic trigonometric identities are:

• sin² 𝜃 + cos² 𝜃 = 1
• 1 + tan² 𝜃 = sec² 𝜃
• 1 + cot² 𝜃 = cosec² 𝜃

Some additional trigonometric formulae:

• sin (A + B) = sin A cos B + cos A sin B
• sin (A – B) = sin A cos B – cos A sin B
• cos (A + B) = cos A cos B – sin A sin B
• cos (A – B) = cos A cos B + sin A sin B
• tan (A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
• sin 2A = 2sin A cos A
• cos 2A = cos² A – sin² A = 1 – 2sin² A = 2cos² A – 1
• tan 2A = 2tan A/(1 – tan²A)

Trigonometric formulae according to different quadrants.

Some formulae for complementary angles:

Also Check: Mathematical Tools for Physics Class 11 (Important Concepts and Problems)

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