Trigonometry is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and angles. Many different formulas are often used with trigonometry to find values of unknown quantities, or simply for solving problems.

Consider a right-angled triangle, which is right angled at B

Therefore, sin A = Perpendicular / Hypotenuse = BC/ AC

Cos A = Base / Hypotenuse = AB / AC

Tan A = Perpendicular / Base = BC / AB = Sin A / Cos A

The Cosecant, Secant and Cotangent Ratios

These three functions sin, cos and tan are the main functions in trigonometry. But there are three other functions that are reciprocal of these functions. They are referred to as reciprocal trigonometrical functions.

These are secant function, cosecant function and cotangent function.

The values of these functions are:

Sec A = 1 / Cos A

Cosec A = 1 / Sin A

Cot A = 1 / Tan A = Cos A / Sin A

Identities

1.     1 + tan^2 A = sec^2 A

2.     Cot^2 A + 1 = Cosec^2 A

3.     Sin^2 A + cos^2 A = 1

·       Proof of 1 + tan^2 A = sec^2 A

Consider a right-angled triangle right angled at B with sides, AB (base), BC (Perpendicular) and AC (hypotenuse)

According to Pythagoras theorem

AC^2 = AB^2 + BC^2

Divide each side by AB^2

AC^2 / AB^2 = 1 + BC^2 / AB^2

Put BC / AC = tan A and AC / BC = sec A

That gives, Sec^2 A = 1 + tan^2 A

Question: How to prove Cot^2 A + 1 = Cosec^2 A?

Answer: This identity can be proved by same method that is by applying Pythagoras theorem. For detailed information visithttps://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ.

Important Points:

·       The function cosec is reciprocal of sine.

·       Whenever the value of the sine function is zero, then cosec function is not defined.

·       Similarly, the secant and cot function do not have defined values when cos and tan function have 0 value respectively.

The Compound Angle 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

·       Tan (A – B) = tan A – tan B / 1 + tan A tan B

Double Angle Formulae

These are derived by using the compound angle formulae:

·       Sin 2A = 2 Sin A cos A

·       Cos 2A = Cos^2 A – sin^2 A = 2 cos^2 A – 1 = 1 – 2 sin^2 A

·       Tan 2A = 2 tan A / 1 – tan^2 A

Further Trigonometric Identities

These identities are formulated by combining the double angle and compound angle formulae.

·       Tan 3x = 3 tan x – tan^3 x / 1 – 3 tan^2 x

·       Cot y – Cot x = 2 sin (x – y) / cos (x – y) – cos (x + y)

Question: How to prove Tan 3x = 3 tan x – tan^3 x / 1 – 3 tan^2 x?

Answer: Proof- tan 3x can be written as

= tan(3x) = tan (2x + x)

 Now we have,

= tan (2x + x) = [tan(2x) + tan(x)] / [1 - tan(2x) tan(x)] (According to compound angle formulae)

= tan(2x + x) = [tan x + {(2 tan x) / (1 - tan2x)}] / [1 - {(2 tan x) / (1 - tan2x)} tan x]

Substitute the value of tan 2x from double angle formula

tan(2x + x) = (tan x - tan3x + 2 tan x) / (1 - tan2x - 2 tan2x) = (3 tan x - tan3x) / (1- 3 tan2x)

Thus, tan3x = (3 tan x - tan3x) / (1- 3 tan2x)

For the derivation of other such formulae visit https://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ. 

Expressing a sin θ ± b cos θ in the for R sin(θ ± α) or R cos(θ ± α)

·       a sin θ ± b cos θ = R sin(θ ± α)

·       a cos θ ± b sin θ = R cos(θ ± α)

where R = (a^2 + b^2)^1/2 and tan α = b / a

Logarithms to Base 10

The study of logarithms begins with logarithms with base 10. Logarithms to base 10 are only commonly used because we use a decimal system of counting.

Rules for writing logarithms of base 10:

·       Log10 x is equal to the power of 10 that must be raised to obtain the general number x.

·       The rule for base 10 is: if y = 10^x then x = log10 y

Logarithm Laws

If x and y are positive and also a > 1 and a not equal to zero, then following laws can be applied to logarithms:

Rule 1: For any real number x, log10 10^x = x.

Rule 2: For any real number x > 0, 10^ (log10 x) = x

Note: Rule 1 and 2 states that y = 10^x and y = log10 x, are both inverse functions of each other.

Rule 3: For any real numbers x > 0 and y > 0, log10 xy = log10 x + log10 y.

Rule 4: For any real numbers x > 0 and y > 0, log10 ( x / y ) = log10 x − log10 y.

Rule 5: For real numbers x and n, with x > 0, log10 xn = n log10 x

Logarithms to Base a

Suppose a > 0 and a not equal to 1 and also x > 1. The logarithm to base a is the function is defined as:

 y = a^x then x = log a (y). The function log a is pronounced “log base a.” It is the inverse of f(x) = a^x.

Laws of Logarithm for log with base a for a > 1

Rule 1: For any real number x, loga ax = x

Rule 2: For any real number x > 0, a loga x = x

Rule 3: For any real numbers x > 0 and y > 0, loga xy = loga x + loga y, this is called multiplication law.

Rule 4: For any real numbers x > 0 and y > 0, logb x y = loga x − loga y, this is called division law.

Rule 5: For real numbers x and n, with x > 0, loga x^n = n loga x, this is called power law

Rule 6: For numbers x > 0, a > 1 and b > 1, loga x = loga b × logx a

According to rule 6, we can see that

logb a × loga b = logb b = 1

and also, logb a = 1/ loga b

Rule 7: loga (1 / x) = -loga x

Question: How to solve logarithmic equations?

Answer: When solving equations with laws, it is essential to analyze all roots in the original equation because loga x only exits for x > 0 and a > 0 and a not equal to 1. To understand better with the help of examples kindly visithttps://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ.

Exponents and Solution of Exponential Equations

Any number which is raised to the power of another number is an exponential function.

For eg. y = f(x) = a^x is an exponential function where a is raised to the power of x and a > 0 and a not equal to 1.

Rules of Exponents:

1.     a^n × a^m = a^ (n + m)

2.     a^n / a^m = a^ (n – m)

3.     (a^m)^n = a^(nm)

4.     a^0 = 1

5.     (a / b)^x = a^x / b^x

Question: How to solve exponential equations?

Answer: Exponential equations can be solved by taking log on both sides and then equating each other with the help of exponential and logarithmic rules to get the required solution. To learn with the help of examples visithttps://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ.

Solving an Exponential inequality

To solve an equality, the same rule is applied that is taking log of both the sides to any suitable base but the sign of the equality must be taken into consideration.

·       If the base is greater than 1 then equality symbol remains the same.

·       If base is between 0 and 1, then sign of the inequality is reversed.

Natural Logarithms

When logarithm is taken as the base e, then it is called natural logarithm. Here e = 2.718 (approx.)

y = e^x is the natural exponential function, and x = ln y

ln x is written to represent loge x

The graphs of e^x and 10^x are very similar to each other when plotted on x and y axis, and therefore their logs behave in same manner.

Transforming an equation to linear form

Relationships of the form y = kx^n and y = k(a^x), where k, a, and n are constants, can be transformed using logarithms into straight lines of the kind Y = mX + c, where X and Y are functions of x and y. 

What is Algebra?

Algebra, which literally means "the science of calculation", is a form of mathematical expression used for solving problems. Algebraic equations are also used in calculus, computer algebra, and various fields such as physics, engineering, and economics.

The Modulus Function

The size of a number, whether it is positive or negative, is known as its modulus. For instance, the modulus of 7 is 7, and the modulus of −7 is also 7.

Points to Remember

·       The modulus of a number is represented as |x|, where x is the number.

·       |x| = x for x>= 0 and |x| = -x for x< 0.

·       If the value of the function f(x)= -2, then |f(x)|= 2

·       If the modulus is contained within the function, such as in f(|x|), then the modulus must be applied to the x-value before the function can be applied, as in f (|-4|) = f (4).

Graphs of Modulus Function

Generally, 3 main types of graphs can be drawn for a modulus function.

·       y=|f(x)|: By reflecting the negative portion of the f(x) graph in the x-axis, all negative values of f(x) are converted to positive values. This limits the possible range to |f(x)|≥0 (or a subset within |f(x)|≥0.

·       y=f(|x|): The graph of f(x) for x≥0 is reflected in the y-axis for negative x-values because negative x-values produce the same outcome as their corresponding positive x-values.

·       y=|f(−x) |: The x-values change signs (from positive to negative or from negative to positive), causing the y-axis to display the graph of f(x). Then, by reflecting the negative portion of the f(x) graph in the x-axis, all negative values of f(x) are converted to positive values. The range is limited, just like with y=|f(x)|.

Question: Can we draw graphs of modulus function that are in the form of quadratic or cubic equations?

Answer: Graphs of cubic and quadratic equations containing modulus function can be drawn with the help of above-mentioned rules. To understand more clearly on this topic, visit https://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ.

Polynomials

Polynomials are algebraic expressions with indeterminates and constants in them. Polynomials can be thought of as a mathematical dialect. They are used to express numbers in almost every field of mathematics and are extremely important in some, such as calculus.

An expression of the form f(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0 is a polynomial. Here, each term has their meanings:

·       x is the variable

·       n is non negative integer

·       the coefficients an, an−1, a2, a1 are constants.

·       an is the leading coefficient and an not equal to 0.

·       a0 is the constant term.

Degree of the polynomial is given by the highest power of x in that polynomial.

Methods that can be used for division of a polynomial are:

1.     Long Division: The regular method as we do in arithmetic operations can be used to divide a polynomial by another polynomial.

2.     Factorization Method: When dividing polynomials, you may need to factor the polynomial in order to find a common factor between the numerator and denominator. The common factor gets canceled out and we get the result.

3.     Splitting Method: The polynomials can be split with the help of arithmetic operators + or - , and then can be further simplified.

Division of a polynomial can be represented as:

P(x) = D(x) * Q(x) + R(x)

Where, P(x) = Dividend

D(x) = Divisor

Q(x) = Quotient

R(x) = Remainder

Question: How the division of a polynomial carried by long division method?

Answer: Division of a polynomial by its divisor is somewhat complicated than the normal division method. To help yourself with polynomial division kindly visithttps://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ.

The Factor Theorem

If f(x) is a polynomial of degree n greater than or equal to 1, and a is any real number, then (x - a) is a factor of f(x) if f(a) = 0. In other words, if f(a) = 0, then (x - a) is a factor of f(x).

Formula:

According to the factor theorem, (y - a) is a factor of the polynomial g(y) of degree n ≥1 if and only if g(a) = 0. In this case, a can be any real number. The factor theorems formula is g(y) = (y - a) q (y). This is valid if

·       (y – a) is factor of g (y).

·       g (a) = 0.

·       When g(y) is divided by, the remainder becomes zero (y – a).

·       When g(y) is divided by, the remainder becomes zero (y – a).

·       a is the solution to g(y) = 0, and a is the zero of the function g(y).

The Remainder Theorem

According to the remainder theorem, when a polynomial p(x) is divided by a linear polynomial (x - a), the remainder equals p(a).

For applying the remainder theorem, the following steps must be followed:

·       Set the linear polynomial to zero to find its zero. x - a = 0, implying that x = a.

·       Then start by putting it into the given polynomial. The remainder would be the result.

·       It is worth noting that the degree of the remainder polynomial is always one less than that of the divisor polynomial. Using this fact, the remainder of any polynomial divided by a linear polynomial (degree 1) must be a constant (whose degree is 0).

Question: Are there proofs to factor theorem and remainder theorem?

Answer: Of Course, factor theorem and remainder theorem can be proved with the help of this videohttps://youtube.com/channel/UCoqI7C9rI2UbFPITF2bPgnQ.