∠ θ	0°	30°	45°	60°	90°
Sin θ	0	½	1/√2	√3/2	1
Cos θ	1	√3 / 2	1/√2	½	0
Tan θ	0	1/√3	1	√3	∞
Cosec θ	∞	2	√2	2/√3	1
Sec θ	1	2/√3	√2	2	∞
Cot θ	∞	√3	1	1/√3	0

Read More about Trigonometric Table.

Some Special Equations

Apart from the equations that are just discussed above, there are some special equations as well, which can be called as equations of shapes, that are discussed further.

Equation of a Line

The general form of the equation of a line in two variables and first degree is Ax+By+C=0

A, B, and C are constant real numbers and a graphical representation of the line equation will give a straight line

The equation of a line in slope intercept form can be written as:

y = mx+c

Where ‘m’ is the slope of the line and ‘c’ is the intercept of y.

The general equation of line can be written in slope intercept form as:

• y = (-A/B)x – (C/B)
• m = (-A/B)
• c = (C/B)

“The intercept is the point through which the line crosse x-axis and y-axis”

Equation of Circle

The general equation of a circle is x² + y²+ 2gx + 2fy + c = 0 , where g, f and c are constants. This form allows us to represent circles with centers not necessarily at the origin and can represent circles with various characteristics. The values of g,f and c determine the center, radius, and other properties of the circle.

• Equation of a Circle with Center at Origin

When the center of a circle is at the origin (0, 0), then the equation will be: x²+y²= r² , where r is the radius of the circle. This equation represents a circle with center at the origin with a radius of r.

• Equation of a Circle with Center Not at Origin

To expre­ss the equation of a circle with ce­nter (h,k) and radius r: (x-h)²+(y-k)² = r². In this equation, the value­s (h,k) represent the­ coordinates of the cente­r, while r represe­nts the radius of the circle. Esse­ntially, this equation defines a circle­ centered at (h,k) with a radius e­qual to r.

Equation of Parabola

A parabola is a curve that looks the same on both sides and has three important parts, namely, the focus, directrix, and axis of symmetry. The focus is in the middle on the axis of symmetry, while the directrix is a parallel line. The parabola is consist of points where the distance from the focus is the same as the distance from the directrix and the sharpest point on the parabola is called the vertex.

The general equation of a parabola is given as:

• Standard Equation for a Regular Parabola

y² = 4ax

• Vertical Parabola

y = a(x-h)²+ k

where h,k represents the vertex.

• Horizontal Parabola

x = a(y-k)²+ h

where h,k represents the vertex.

Equation of Ellipse

An ellipse­ can be define­d as the set of points where­ the sum of the distances from any point on the­ shape to two fixed points (known as foci) remains constant.

A circle is also an ellipse with it’s foci at the same point. Ellipse can be defined by it’s two axis x-axis and y-axis called as mazor axis and minor axis.

The general form of ellipse equation is:

(x-h)²/a² + (y-k)²/b² = 1

Where,

• (h, k) are the coordinates of the center of the ellipse.
• The se­mi-major axis, denoted as ‘a’ repre­sents the distance from the­ center of an ellipse­ to its furthest point along the major axis.
• The se­mi-minor axis, marked as ‘b’, refers to the­ distance from the cente­r of the ellipse to its farthe­st point along the minor axis.

Equation of Hyperbola

A hyperbola be­longs to a group of curves called conic sections. The­ equation that describes a hype­rbola depends on its orientation and the­ location of its center.

Here­ are the standard form equations for hype­rbolas:

• Horizontal Hyperbola

For a horizontal hyperbola with its ce­nter at the point (h, k), a horizontal major axis, and a vertical minor axis, the­ equation is as follows:

(x-h)²/a² – (y-k)²/b² = 1

Where,

• (h, k) is the the middle of the hyperbola.
• a shows how far it is from the middle to a point on the longer line.
• b shows how far it is from the middle to a point on the shorter line.

• Vertical Hyperbola

The e­quation for a vertical hyperbola with its cente­r at the point (h, k), a vertical major axis, and a horizontal minor axis can be e­xpressed as:

(y-k)²/a² – (x-h)²/b² = 1

Where,

• (h, k) is the center of the hyperbola.
• a shows distance from the center to a vertex along the longer side.
• b shows distance from the center to a vertex along the shorter side.

Equation in Calculus

There are two branches of Calculus namely, differential calculus and integral calculus which deals with finding derivative and antiderivative of a function respectively. The equation in differential and integral calculus are discussed below:

Differential Equation

A differential equation involves terms and derivatives of a variable (dependent) with respect to another (independent). For instance, dy/dx = f(x), where x is independent and y is dependent. It represents a rate of change and describes how one quantity changes in relation to another. Various formulas exist for solving these equations.

The order of a differential equation is determined by the highest derivative present. For example, dy/dx = 7x + 8 is a first-order equation, while (d²y/dx²) + 2(dy/dx) + y = 0 is a second-order one. First-order equations involve only the first derivative, like dy/dx, while second-order equations include the second derivative, like d²y/dx².

The degree of a differential equation is the highest power of the derivative it contains. It’s determined by looking at the derivatives in the equation. For example, if an equation has a second derivative like (d²y/dx²), it’s a first-degree equation. Some equations may not have a defined degree if they don’t fit a polynomial pattern.

Integral Equations

Integration is like adding up specific pieces discrete data. It’s used to find functions that describe area, displacement, or volume, which are made up of many small pieces that can’t be measured separately.

In math, calculus uses a concept called “limits” to get accurate answer. Limits are a way to look at what happens to the points on a graph as they come closer until the distance between two points are equal to zero.

There are basically two problems where integration is used:

• To find the original function when we have its rate of change.
• To figure out the area under a function’s graph within specific rules..

Integral Calculus: Integral calculus is all about these two problems. It has two parts: definite (specific area) and indefinite (finding the original function). The Fundamental Theorem of Calculus connects differentiation and integration.

Definite Integral: An integral that contains the upper and lower limits then it is a definite integral. On a real line, x is restricted to lie. Riemann Integral is the other name of the Definite Integral. A definite Integral is represented as

[Tex]\int_a^bf(x)dx = F(b) – F(a) [/Tex]

Where,

F is the antiderivative of ‘f’ {dF(x)/dx = f(x)}

Indefinite Integral: An indefinite integral doesn’t have these upper and lower limits. It’s written like this:

∫f(x)dx = F(x) + C

In this equation, C is any constant, and f(x) is the function we’re integrating, also known as the integrand. It means finding the function that, when you take its derivative, gives you f(x).

Systems of Equations

Systems of e­quations are mathematical expre­ssions that involve multiple variables. The­ goal is to find values for these variable­s that satisfy all the given equations at the­ same time. There are­ different methods to solve system of linear equations, such as substitution, elimination, and graphical represe­ntation. The outcomes can include a single­ solution, no solution when lines are paralle­l, or an infinite number of solutions when line­s coincide

System of Linear Equations

A system of linear equations is a set of two or more equations in which the variables have exponents of 1 and are related linearly, meaning they form straight lines when graphed. The goal is to find values of the variables that satisfy all the equations in the system simultaneously.

System of Non-Linear Equations

A system of non-line­ar equations consists of two or more equations whe­re the relationships be­tween variables are­ not linear. These re­lationships involve variables raised to powe­rs other than 1, such as squares, cubes, or othe­r nonlinear functions. Solving these syste­ms can be more complex compare­d to linear systems because­ nonlinear equations can produce curve­s instead of straight lines when graphe­d.

Example of System of Non-Linear Equations

• Equation 1: x² + y = 8
• Equation 2: xy + 2 = 0

In this system, x and y are the variables, and the equations involve nonlinear terms, such as x² and xy. Solving such systems typically requires advanced techniques and may not always have analytical solutions.

How to Solve Systems of Equations?

There are 3 ways of solving systems of equations- Graphical method, substitution method and elimination method

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can be solved.

• Equation (1): 2x + 3y = 8
• Equation (2): 3x -2y = 4

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

Step 1: Solve any one equation for x

2x+3y=8

⇒ 2x =-3y+8 …….(subtracting both sides by 3)

⇒ x = ½ (-3y+8) ………(dividing both sides by 2)

⇒ x = -3/2y+4 ……… equation (3)

Step 2: Substitute the value of x that we just got in the other equation 3x – 2y = 4

3 [(-3/2)y+ 4] – 2y= 4

⇒ (-9/2)y + 12 – 2y= 4

⇒ (-13/2)y + 12 = 4 ……….(adding the coefficients of y)

⇒ (-13/2)y= -8 ………..(subtracting 12 from both sides)

⇒ y = 16/13 (dividing both sides by -13/2 or multiplying by the reciprocal of this fraction -2/13)

Step 3: Substitute this value of y in equation (3)

x= -3/4 × [16/13 + 4]

⇒ x = -24/13 +4

⇒ x= 28/13

Now the system is solved and we have value of x and y as

x = 28/13 and y= 16/13

Graphical Method

The graphical method involves graphing each equation on the same coordinate plane and finding the point(s) where the graphs intersect. These points represent the solutions to the system of equations.

For the given

• Equation (1): x + 2y=8
• Equation (2): 3x – 2y=4

Plot both the equations on the graph, the point where both the equations intersect is the solution. Putting the values of x and y we can find that the graph of two lines intersect at (3, 5/2) which is the required solution of the graph.

Learn More, Graph of Linear Equations in Two Variables

Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable. This creates a new equation with only one variable, which can be solved.

Taking the same equation to solve using elimination method

• Equation (1) : 2x + 3y = 8
• Equation (2) : 3x – 2y = 4

Step 1: To make the coefficients of variable x in both equations equal, multiply equation (1) by 3 and equation (2) by 2. The equation formed are

6x + 9y = 24

⇒ 6x – 4y = 8

Step 2: Subtract like terms in equation (2) from equation (1)

6x – 6x + 9y + 4y = 24 – 8

⇒ 13y = 16

⇒ y = 16/13

Step 3: Put the value of y in equation (2)

3x – 2(16/13) = 4

⇒ 3x – (32/13) = 4

⇒ 3x = 4+ (32/13)

⇒ 3x = 84/13

⇒ x = 28/13

Hence, the value of x and y are

x = 28/13 and y= 16/13 

Solving Equations

Solving equations involve­s finding the values that make a give­n equation true. There­ are various techniques for solving diffe­rent types of equations, such as line­ar, quadratic, polynomial, exponential, and trigonometric e­quations. Each type of equation require­s its own specific method or function to find the solutions, providing a range­ of versatile methods for tackling mathe­matical problems.

Solving Linear Equations

Example: Check if 5 = 2x +3

Solution:

Put x = 1

5 = 2(1) + 3

⇒ 5 = 2+3

⇒ 5 = 5

LHS = RHS

Solving Quadratic Equations

There are various ways of solving a quadratic equation- factor method, completing squares method and quadratic formula.

The quadratic formula to find variable x in a quadratic equation ax² + bx + c = 0 can be written as:

[Tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a} [/Tex]

Let us solve a quadratic equation x²-3x=28 using the quadratic formula

Step 1: convert the given equation into the standard quadratic equation ax²+bx+c=0

x²– 3x – 28=0

Here, a = 1, b = -3, c = -28

Step 2: Put the values of a,b, and c in the quadratic formula

[Tex]\frac{-\left(-3\right)\pm\sqrt{\left(-3\right)^2-4\left(-28\right)}}{2} [/Tex]

[Tex]\Rightarrow \frac{-\left(-3\right)\pm\sqrt{9-4\left(-28\right)}}{2}  [/Tex]………..(solving  [-3²])

[Tex]\Rightarrow \frac{-\left(-3\right)\pm\sqrt{9+112}}{2}  [/Tex]………….(multiplying -4 by -28)

[Tex]\Rightarrow \frac{-\left(-3\right)\pm\sqrt{121}}{2}  [/Tex]…………..(adding values in root)

[Tex]\Rightarrow \frac{-\left(-3\right)\pm11}{2}  [/Tex]…………(solving root)

[Tex]\Rightarrow \frac{\left(3\right)\pm11}{2}  [/Tex]……….. ( – multiplying with minus is plus)

Now solve the equation [Tex]x=\frac{\left(3\right)\pm11}{2}  [/Tex]when ± is plus. Add 3 to 11.

x= 14/2

⇒ x = 7

Now solve the equation [Tex]x=\frac{\left(3\right)\pm11}{2}  [/Tex]when ± is minus. Subtract 11 from 3.

x =(-28)

⇒ x =(-4)

Hence x = 7, -4

Solving Polynomial Equations

There are various ways of solving a polynomial equation based on its complexity. The main methods to solve a polynomial equations are:

Roots of a Polynomial: A root of a polynomial is a value of x that makes polynomial equivalent to zero, i.e, P(x)=0

Factorization and Division: To find the roots of a mathe­matical expression, the common technique used is called factorization of polynomial. This me­thod involves breaking down a complex e­xpression into simpler components that are­ more easy to solve­. For instance, if there is an e­quation like x²-4, by dividing the equation into (x-2)(x+2) and then solve­ for x by equating each part to zero: x-2=0 and x+2=0 . The­ values of x that make each compone­nt zero correspond to the roots of the­ equation.

Complex Roots: Some polynomials, such as x²+1 , do not have real number roots. However, by considering complex numbers, every non-constant polynomial has at least one root. This is known as the fundamental theorem of algebra.

Solving Exponential Equations

Exponential Equations looks complex which requires basic algebra knowledge. In mathematics there are mainly three types of exponential equations to solve and hence there are three methods to solve them, namely:

• Equations with Exponents having Same Base

Let’s take an equation 10^2+x = 10³ to understand more clearly.

In this equation 10 is the base on both side which is common. So, to solve this equation ignore the base and solve for x.

2+x=3

⇒ 2-2+x = 3-2……….(substracting 2 from both sides)

⇒ x=1

Now, put this value of x in the original equation

10^2+x = 10³

⇒ 10³ = 10³

Hence solved.

• Equations with Exponents and Whole Numbers

Suppose, 2^x+5 – 2 = 14

In this equation there are whole numbers on both the side.

∴ solve the whole numbers

2^x+5 -2+2 = 14+2 ………… (adding 2 on both sides)

⇒ 2^x+5 = 16

Covert 16 into an exponent with the base 2

⇒ 2×2×2×2 = 16

∴ 2⁴ = 16

Put 2⁴ in place of 16

2^x+5 = 2⁴

Now that the base are same, so ignore base and solve for x

x + 5 = 4

⇒ x = (-1)

Put the value of x in the original equation

2^(-1)+5 – 2 = 14

⇒ 2⁴ – 2 = 14

⇒ 16-2 = 14

⇒ 14 = 14

Hence, solved.

• Using Log terms without Same Base

In the equation 2^x = 4^y, (2 and 4) are the base, which are not same.

To find the value of x and y , multiply the equation on both side with natural logarithm (In)

ln(2^x ) = ln(4^y )

Using the Power rule of logarithm, we will get

x ln(2) = y ln(4)

Solve for y in terms of x

y = xln(2)/ln(4)

this gives us a relationship between x and y

Solving Trigonometric Equations

Trigonometric e­quations contain expressions with trigonometric functions such as sine­, cosine, and tangent. To find the solutions for a variable­ x within the range 0 ≤ x ≤ 2π, we ide­ntify the principal solutions. If these solutions incorporate­ an integer ‘n,’ they are­ known as general solutions.

The table below shows the general trigonometric equations with their general solutions.