Essential Maths Formulas for CBSE Class 10: Your Ultimate Guide
Mathematics in Class 10 is a pivotal phase for students in the CBSE curriculum, acting as a foundation for advanced concepts in higher grades. At this stage, formulas in mathematics are not just numerical expressions but powerful tools that unlock the potential to solve a wide range of problems. They serve as the backbone of mathematical reasoning and problem-solving.
The importance of formulas in Class 10 Maths lies in their ability to simplify complex problems into manageable steps. These formulas are like shortcuts that guide students through the path of algebra, geometry, trigonometry, and other critical areas of mathematics. By mastering these formulas, students can approach mathematical problems with confidence and precision.
Understanding and memorizing these formulas is crucial, but the real mastery is achieved when students learn to apply them in various contexts. This application enhances analytical thinking, promotes logical reasoning, and fosters a deeper understanding of mathematical concepts. It’s not just about finding the right answer; it’s about developing a mindset that methodically dissects problems and finds efficient solutions.
Moreover, these formulas are not confined to the classroom. They have real-world applications, making their understanding essential for practical life. Be it in engineering, economics, or everyday calculations, the formulas learned in Class 10 Maths have a far-reaching impact.
Mastering Class 10 Maths formulas is a stepping stone to academic success in mathematics and beyond. It empowers students with the skills to tackle complex problems, not just in exams but in real-life scenarios, fostering a lifelong appreciation for the beauty and utility of mathematics.
Let us now look at the chapter wise CBSE Class 10 Math formulas.
1. Algebra
In Class 10 CBSE Maths, Algebra is key, covering Linear Equations, Quadratic Equations, and Arithmetic Progression. Understanding formulas like axÂ² + bx + c = 0 and Tn = a + (n-1)d enhances problem-solving and logical reasoning, preparing students for advanced studies and real-world applications.
Algebraic formulas
$$ a^2 – b^2 = (a – b)(a + b) $$
$$ (a + b)^2 = a^2 + 2ab + b^2 $$
$$ a^2 + b^2 = (a + b)^2 – 2ab $$
$$ (a – b)^2 = a^2 – 2ab + b^2 $$
$$ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca $$
$$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3; \quad (a + b)^3 = a^3 + b^3 + 3ab(a + b)$$
$$(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3; \quad (a – b)^3 = a^3 – b^3 – 3ab(a – b)$$
$$a^3 + b^3 = (a + b)(a^2 – ab + b^2)$$
$$a^3 – b^3 = (a – b)(a^2 + ab + b^2)$$
$$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$$
$$(a – b)^4 = a^4 – 4a^3b + 6a^2b^2 – 4ab^3 + b^4$$
$$a^4 – b^4 = (a – b)(a + b)(a^2 + b^2)$$
$$a^5 – b^5 = (a – b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$$
$$\text{If } n \text{ is a natural number } a^n – b^n = (a – b)(a^{n-1} + a^{n-2}b + \dotsb + ab^{n-2} + b^{n-1})$$
$$\text{If } n \text{ is even } (n = 2k), a^n + b^n = (a + b)(a^{n-1} – a^{n-2}b + \dotsb + ab^{n-2} – b^{n-1})$$
$$\text{If } n \text{ is odd } (n = 2k + 1), a^n + b^n = (a + b)(a^{n-1} – a^{n-2}b + \dotsb – ab^{n-2} + b^{n-1})$$
$$(a + b + c + \dotsm)^2 = a^2 + b^2 + c^2 + \dotsm + 2(ab + ac + bc + \dotsm)$$
$$\text{Laws of Exponents } (a^m)^n = a^{mn}; \quad (ab)^m = a^m b^m; \quad (a^m)^n = a^{mn}$$
Basic formulas for powers
$$ p^m \times p^n = p^{m+n} $$
$$ \frac{p^m}{p^n} = p^{m-n} $$
$$ (p^m)^n = p^{mn} $$
$$ p^{-m} = \frac{1}{p^m} $$
$$ p^1 = p $$
$$ p^0 = 1 $$
Arithmetic Progression(AP) Formulas
If a1, a2, a3, a4, a5, a6,â€¦ are the terms of AP and d is the common difference between each term, then we can write the sequence as; a, a+d, a+2d, a+3d, a+4d, a+5d,â€¦.,nth termâ€¦ where a is the first term. Now, nth term for arithmetic progression is given as;
$$ n^{\text{th}} \text{ term} = a + (n-1)d $$
Sum of the first n terms in Arithmetic Progression;
$$ S_n = \frac{n}{2} [2a + (n – 1)d] $$
Linear Equations
One Variable | ax+b=0 | aâ‰ 0 and a&b are real numbers |
Two variable | ax+by+c = 0 | aâ‰ 0 & bâ‰ 0 and a,b & c are real numbers |
Three Variable | ax+by+cz+d=0 | aâ‰ 0 , bâ‰ 0, câ‰ 0 and a,b,c,d are real numbers |
Pair of Linear Equations in two variables
$$ a^1 x + b^1 y + c^1 = 0 $$
$$ a^2 x + b^2 y + c^2 = 0 $$
Where a1, b1, c1, a2, b2, and c2 are all real numbers and $$a_1^2 + b_1^2 \neq 0$$ $$a_2^2 + b_2^2 \neq 0$$ It should be noted that linear equations in two variables can also be represented in graphical form.
Quadratic Equation and Formula
The standard form of a Quadratic Equation is:
$$ ax^2 + bx + c = 0 \text{ where } a \neq 0 \text{ and,} $$
$$ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} $$
Summarized Table:
Topic | Formula Name | Formula | Denotation |
Linear Equations | Solution of Linear Equations | ax+b=0 | aâ‰ 0 and a&b are real numbers |
Quadratic Equations | Standard Form of a Quadratic Equation | $$ ax^2 + bx + c = 0$$ | a, b, c are coefficients |
Quadratic Formula | $$ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} $$â€‹ | x is the variable, a, b, c are coefficients | |
Arithmetic Progression (AP) | nth Term of AP | $$ n^{\text{th}} \text{ term} = a + (n-1)d $$ | Tn is the nth term, a is the first term, d is the common difference, n is the term number |
Sum of First n Terms of AP | $$ S_n = \frac{n}{2} [2a + (n – 1)d] $$ | Sn is the sum of the first n terms, a is the first term, d is the common difference, n is the number of terms |
2. Geometry
Geometry in Class 10 Maths encompasses the study of shapes, sizes, and properties of space. Key topics include Triangles, Circles, and Coordinate Geometry. Mastering these formulas, like the Pythagorean Theorem or the Circle Area formula, is crucial for solving various geometrical problems effectively.
Geometry Shapes Formulas for Class 10 | |
Name | Formula |
Area of Triangle | $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$ |
Pythagorean Theorem | $$ a^2 + b^2 = c^2 $$ |
Area of a Circle | $$ \text{Area} = \pi r^2 $$ |
Circumference of a Circle | $$ C = 2\pi r \text{ or } \pi d $$ |
Area of a Parallelogram | $$ \text{Area} = \text{base} \times \text{height} $$ |
Area of a Trapezoid | $$ \text{Area} = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height} $$ |
Area of a Kite or a Rhombus | $$ \text{Area} = \frac{1}{2} \times (\text{diagonal}_1 + \text{diagonal}_2) $$ |
Area of a Square | $$ \text{Area} = \text{side}^2 $$ |
Area of a Regular Polygon | $$ \text{Area} = \frac{1}{2} \times \text{perimeter} \times \text{apothem} $$ |
Number of Diagonal in n-sided Polygon | $$ \text{Diagonals} = \frac{1}{2} n(n-3) $$ |
Slope | $$ m = \frac{(y_2 – y_1)}{(x_2 – x_1)} = \frac{\text{rise}}{\text{run}} $$ |
Midpoint Formula | $$ (x_{\text{mid}}, y_{\text{mid}}) = \left(\frac{(x_1+x_2)}{2}, \frac{(y_1+y_2)}{2}\right) $$ |
Distance Formula | $$ d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} $$ |
Equation of a Circle | $$ (x – h)^2 + (y – k)^2 = r^2 $$ |
3. Trigonometry
Trigonometry in Class 10 forms a crucial part of the mathematics curriculum, guiding students through the intriguing world of angles and sides of triangles, especially right-angled triangles. This branch of mathematics introduces various ratios, identities, and functions that are foundational in solving complex geometrical problems. Class 10 trigonometry extends beyond theoretical knowledge, finding practical applications in diverse fields such as engineering, astronomy, physics, and architecture. The understanding of trigonometry paves the way for more advanced mathematical concepts like Linear Algebra, Calculus, and Statistics, making it an indispensable tool for students aspiring to excel in mathematics and its applications
Trigonometry is a novel and challenging topic introduced in the CBSE Class 10 curriculum, requiring students to learn and effectively apply a variety of formulas. This chapter focuses on the relationships between the angles and sides of right-angled triangles, making it essential for students to grasp these concepts for solving trigonometry-related problems.
Below is a list of crucial Trigonometric Formulas for Class 10 students. These formulas, which are fundamental for understanding and solving trigonometric problems, are readily accessible here for learning and reference.
In trigonometry, the formulas primarily revolve around the three sides of a right-angled triangle: the adjacent side (or base), the perpendicular, and the hypotenuse. By applying the Pythagorean theorem to a right-angled triangle, we establish the relationship:
$$ (\text{Perpendicular})^2 + (\text{Base})^2 = (\text{Hypotenuse})^2 $$
Or, in shorter terms:
$$ P^2 + B^2 = H^2 $$
We will now explore the formulas derived from trigonometric ratios, including sine, cosine, tangent, secant, cosecant, and cotangent. These formulas are integral to understanding and solving problems in trigonometry, especially in the context of Class 10 Maths.
Property | Mathematical value |
sin A | Perpendicular/Hypotenuse |
cos A | Base/Hypotenuse |
tan A | Perpendicular/Base |
cot A | Base/Perpendicular |
cosec A | Hypotenuse/Perpendicular |
sec A | Hypotenuse/Base |
Here is an effective way to remember the trigonometric ratios by using a mnemonic phrase where each word’s first letter corresponds to a trigonometric function, and the order of words aligns with the ratios. Here’s a creative and memorable phrase:
“Some People Have, Curly Brown Hair, Turned Permanently Black”
This corresponds to:
- Some People HaveÂ
- Sin = Perpendicular/Hypotenuse |Â
- Cosec (reciprocal of Sin) = Hypotenuse/Perpendicular
- Curly Brown Hair
- Cos = Base/Hypotenuse
- Sec (reciprocal of Cos) = Hypotenuse/Base
- Turned Permanently Black
- Tan = Perpendicular/Base
- Cot (reciprocal of Tan) = Base/Perpendicular
By using this phrase, you can more easily recall the ratios for Sin, Cos, Tan and their reciprocals, Cosec, Sec, and Cot. This mnemonic is particularly useful because it forms a more vivid and therefore memorable sentence, aiding in recall, especially during exams or while solving problems.
Trigonometric Tables
The trigonometric table, especially for a Class 10 student, generally includes the values of trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for standard angles like 0Â°, 30Â°, 45Â°, 60Â°, and 90Â°. Here’s the table:
Angle (Â°) | Sin | Cos | Tan | Cosec | Sec | Cot |
0 | 0 | 1 | 0 | âˆž | 1 | âˆž |
30 | 1/2 | âˆš3/2 | 1/âˆš3 | 2 | 2âˆš3/3 | âˆš3 |
45 | âˆš2/2 | âˆš2/2 | 1 | âˆš2 | âˆš2 | 1 |
60 | âˆš3/2 | 1/2 | âˆš3 | 2âˆš3/3 | 2 | 1/âˆš3 |
90 | 1 | 0 | âˆž | 1 | âˆž | 0 |
An easy and effective way to remember the trigonometric values for 0Â°, 30Â°, 45Â°, 60Â°, and 90Â° is to use a pattern-based approach. Here’s a simple method:
- For Sine (Sin) and Cosine (Cos):
- Remember the sequence: $$ \frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2} $$
- For Sine: Apply the sequence to the angles 0Â°, 30Â°, 45Â°, 60Â°, 90Â° in order.
- For Cosine: Apply the sequence in reverse order to the angles 0Â°, 30Â°, 45Â°, 60Â°, 90Â°.
- For Tangent (Tan):
- Use the formula: Tan = Sin/Cos.
- Calculate: Divide the sine value by the cosine value for each corresponding angle.
- For Cosecant (Cosec), Secant (Sec), and Cotangent (Cot):
- They are reciprocals: Cosec = 1/Sin, Sec = 1/Cos, and Cot = 1/Tan.
- Just invert the values: For each angle, take the reciprocal of Sin, Cos, and Tan respectively.
By remembering the square root sequence and the reciprocal relationships, you can easily reconstruct the entire trigonometric table for these key angles. This method reduces the need to memorize each value individually and instead focuses on understanding a pattern, making it a more efficient and effective way to remember the trigonometric values
Reciprocal Relation Between Trigonometric Ratios
Identity | Relation |
tan A | $$ \frac{\sin A}{\cos A} $$ |
cot A | $$ \frac{\cos A}{\sin A} $$ |
cosec A | $$ \frac{1}{\sin A} $$ |
sec A | $$ \frac{1}{\cos A} $$ |
Trigonometric Sign Functions
$$ \sin(-\theta) = -\sin\theta $$
$$ \cos(-\theta) = \cos\theta $$
$$ \tan(-\theta) = -\tan\theta $$
$$ \csc(-\theta) = -\csc\theta $$
$$ \sec(-\theta) = \sec\theta $$
$$ \cot(-\theta) = -\cot\theta $$
Trigonometric Identities
$$ \sin^2 A + \cos^2 A = 1 $$
$$ \tan^2 A + 1 = \sec^2 A $$
$$ \cot^2 A + 1 = \csc^2 A $$
Periodic Identities
$$ \sin(2n\pi + \theta) = \sin\theta $$
$$ \cos(2n\pi + \theta) = \cos\theta $$
$$ \tan(2n\pi + \theta) = \tan\theta $$
$$ \cot(2n\pi + \theta) = \cot\theta $$
$$ \sec(2n\pi + \theta) = \sec\theta $$
$$ \csc(2n\pi + \theta) = \csc\theta $$
Complementary Ratios
Quadrant I
$$ \sin\left(\frac{\pi}{2} – \theta\right) = \cos\theta $$
$$ \cos\left(\frac{\pi}{2} – \theta\right) = \sin\theta $$
$$ \tan\left(\frac{\pi}{2} – \theta\right) = \cot\theta $$
$$ \cot\left(\frac{\pi}{2} – \theta\right) = \tan\theta $$
$$ \sec\left(\frac{\pi}{2} – \theta\right) = \csc\theta $$
$$ \csc\left(\frac{\pi}{2} – \theta\right) = \sec\theta $$
Quadrant II
$$ \sin(\pi – \theta) = \sin\theta $$
$$ \cos(\pi – \theta) = -\cos\theta $$
$$ \tan(\pi – \theta) = -\tan\theta $$
$$ \cot(\pi – \theta) = -\cot\theta $$
$$ \sec(\pi – \theta) = -\sec\theta $$
$$ \csc(\pi – \theta) = -\csc\theta $$
Quadrant III
$$ \sin(\pi + \theta) = -\sin\theta $$
$$ \cos(\pi + \theta) = -\cos\theta $$
$$ \tan(\pi + \theta) = \tan\theta $$
$$ \cot(\pi + \theta) = \cot\theta $$
$$ \sec(\pi + \theta) = \sec\theta $$
$$ \csc(\pi + \theta) = -\csc\theta $$
Quadrant IV
$$ \sin(2\pi – \theta) = -\sin\theta $$
$$ \cos(2\pi – \theta) = \cos\theta $$
$$ \tan(2\pi – \theta) = -\tan\theta $$
$$ \cot(2\pi – \theta) = -\cot\theta $$
$$ \sec(2\pi – \theta) = \sec\theta $$
$$ \csc(2\pi – \theta) = -\csc\theta $$
Sum and Difference ofÂ Two Angles, Double Angle Formulas, Triple Angle Formulas:
$$ \sin(A + B) = \sin A \cos B + \cos A \sin B $$
$$ \sin(A – B) = \sin A \cos B – \cos A \sin B $$
$$ \cos(A + B) = \cos A \cos B – \sin A \sin B $$
$$ \cos(A – B) = \cos A \cos B + \sin A \sin B $$
$$ \tan(A + B) = \frac{\tan A + \tan B}{1 – \tan A \tan B} $$
$$ \tan(A – B) = \frac{\tan A – \tan B}{1 + \tan A \tan B} $$
$$ \sin 2A = 2 \sin A \cos A = \frac{2 \tan A}{1 + \tan^2 A} $$
$$ \cos 2A = \cos^2 A – \sin^2 A = 1 – 2 \sin^2 A = 2 \cos^2 A – 1 = \frac{1 – \tan^2 A}{1 + \tan^2 A} $$
$$ \tan 2A = \frac{2 \tan A}{1 – \tan^2 A} $$
$$ \sin 3A = 3 \sin A – 4 \sin^3 A $$
$$ \cos 3A = 4 \cos^3 A – 3 \cos A $$
$$ \tan 3A = \frac{3 \tan A – \tan^3 A}{1 – 3 \tan^2 A} $$
4. Mensuration
Mensuration Class 10 Formulas for 2D Figures
Mensuration, a vital component of Class 10 mathematics, delves into the measurement of various geometrical shapes and figures. While the foundation of menstruation is laid in earlier classes, Class 10 significantly expands on these concepts, focusing on the calculation of surface areas and volumes of complex shapes. This chapter consolidates and extends the understanding of these formulas, which are not only crucial for academic success but also have practical applications in fields like architecture, engineering, and daily life calculations. Understanding these mensuration formulas is essential for students as they form the basis for more advanced mathematical studies.
Shape Name | Area | Perimeter (or Circumference) |
Circle | $$ \pi r^2 $$ | $$ 2\pi r $$ |
Square | $$ a^2 $$ | $$ 4a $$ |
Rectangle | $$ l \times b $$ | $$ 2(l + b) $$ |
Triangle | $$ \frac{1}{2} \times \text{height} \times \text{base} $$ | $$ a + b + c $$ |
Parallelogram | $$ b \times h $$ | $$ 2(l + b) $$ |
Rhombus | $$ \frac{1}{2} \times d_1 \times d_2 $$ | $$ 4 \times \text{side} $$ |
Trapezium | $$ \frac{1}{2} \times h \times (a + b) $$ | $$ a + b + c + d $$ |
Mensuration Class 10 Formulas for 3D Figures
Shape Name | TSA | LSA (or CSA) | Volume |
Cube | $$ 6a^2 $$ | $$ 4a^2 $$ | $$ a^3 $$ |
Cuboid | $$ 2 (lb + bh + hl) $$ | $$ 2h (l + b) $$ | $$ l \times b \times h $$ |
Cone | $$ \pi r (r + l) $$ | $$ \pi rl $$ | $$ \left(\frac{1}{3}\right) \pi r^2 h $$ |
Cylinder | $$ 2\pi rh + 2\pi r^2 $$ | $$ 2\pi rh $$ | $$ \pi r^2 h $$ |
Sphere | $$ 4\pi r^2 $$ | $$ 4\pi r^2 $$ | $$ \left(\frac{4}{3}\right) \pi r^3 $$ |
Hemisphere | $$ 3\pi r^2 $$ | $$ 2\pi r^2 $$ | $$ \left(\frac{2}{3}\right) \pi r^3 $$ |
5. Statistics and Probability
Mean
Mean is the most commonly used measure of central tendency. It actually represents the average of the given collection of data. It is applicable for both continuous and discrete data.
It is equal to the sum of all the values in the collection of data divided by the total number of values.
Suppose we have n values in a set of data namely as x1, x2, x3, â€¦, xn, then the mean of data is given by:
$$ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} $$
It can also be denoted as:
$$ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} $$
For grouped data, we can calculate the mean using three different methods of formula.
Direct method | Assumed mean method | Step deviation method |
Mean
$$ \bar{x} = \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i} $$ Here, âˆ‘fi = Sum of all frequencies |
Mean
$$ \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} $$ Here, a = Assumed mean |
Mean
$$ \bar{x} = a + h \frac{\sum f_i u_i}{\sum f_i} $$ Here, a = Assumed mean |
Median
Generally median represents the mid-value of the given set of data when arranged in a particular order. Given that the data collection is arranged in ascending or descending order, the following method is applied:
- If the number of values or observations in the given data is odd, then the median is given by [(n+1)/2]th observation.
- If in the given data set, the number of values or observations is even, then the median is given by the average of (n/2)th Â andÂ [(n/2) +1]th observation.
The median for grouped data can be calculated using the formula,
$$ \text{Median} = l + \left( \frac{\frac{N}{2} – cf}{f} \right) \times h $$
Mode
The most frequent number occurring in the data set is known as the mode.
Consider the following data set which represents the marks obtained by different students in a subject.
Name | Rohan | Aarav | Priya | Deepak | Kirti | Manisha |
Marks Obtained (out of 100) | 73 | 80 | 73 | 70 | 73 | 65 |
The maximum frequency observation is 73 ( as three students scored 73 marks), so the mode of the given data collection is 73.
We can calculate the mode for grouped data using the below formula:
$$ \text{Mode} = l + \left( \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \right) \times h $$
Stats & ProbabilityÂ
Topic | Formula Name | Formula | Denotation |
Statistics | Mean | $$ \frac{\sum \text{ of all values}}{\text{Number of values}} $$ | Summation of all data points divided by the number of points |
Median | Middle value in an ordered list | Data arranged in ascending/descending order and the middle value is chosen | |
Mode | Most frequently occurring value | The value that appears most often in the data set | |
Probability | Basic Probability | $$ \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} $$ | Ratio of the number of favorable outcomes to the total number of possible outcomes |
Conclusion
The mastery of mathematical formulas in Class 10 is not just about memorizing equations; it’s about understanding the language of mathematics. These formulas are the building blocks for higher mathematical concepts and practical applications in various fields like engineering, physics, economics, and even everyday problem-solving. They provide a framework for logical reasoning, analytical thinking, and precision in calculations, which are invaluable skills in both academic and real-world scenarios.
Regular practice of these formulas is essential. Like learning a musical instrument or a new language, proficiency in mathematics comes from consistent practice and application. By routinely solving problems, students reinforce their understanding and develop agility in switching between different concepts. This regular engagement with mathematical formulas also reduces anxiety and builds confidence, preparing students for higher studies and competitive exams.
Additional Resources
To further enhance understanding, students should engage with a variety of resources. Sample problems and solutions, especially those provided in NCERT textbooks, are a great starting point. These examples often cover a wide range of scenarios and problem types.
Previous Year CBSE Math Question PapersÂ
For further reading, reference books like R.D. Sharma and R.S. Aggarwal offer comprehensive coverage of Class 10 mathematics with a multitude of practice problems. Online platforms like Khan Academy provide interactive learning experiences, while educational YouTube channels can offer conceptual explanations and problem-solving techniques.
In addition, participating in mathematics forums and discussion groups can provide insights into different problem-solving methods and expose students to a broader range of questions. These resources, combined with regular classroom learning, can significantly enhance a student’s mathematical proficiency and prepare them for future academic challenges.
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