Mathematics can be a challenging subject, but with the right tricks and techniques, it can become easier and more enjoyable.
In this blog post, we will explore over 100 trending mathematical tricks for students that can help high school learners improve their skills and solve problems more efficiently.
Whether you’re preparing for a test or just want to enhance your math abilities, these tricks will surely come in handy. Let’s dive in!
Tricks for Mental Calculation
1. Multiplying by 11
When multiplying a two-digit number by 11, simply add the two digits together and place the result in the middle. For example, 23 multiplied by 11 is 253 (2 + 3 = 5).
2. Squaring Numbers Ending in 5
To square a number ending in 5, multiply the number before 5 by itself and add 25 at the end. For example, 35 squared is 1225 (3 x 3 = 9, add 25 at the end).
3. Multiplying by 5
When multiplying a number by 5, divide the number by 2 and add a 0 at the end. For example, 36 multiplied by 5 is 180 (36 ÷ 2 = 18, add 0 at the end).
4. Multiplying by 9
When multiplying a number by 9, multiply the number by 10 and subtract the original number. For example, 7 multiplied by 9 is 63 (7 x 10 = 70, subtract 7).
5. Multiplying by 4
When multiplying a number by 4, double the number and double it again. For example, 12 multiplied by 4 is 48 (12 x 2 = 24, double again).
6. Square of Any Number
To square any number, multiply the number by itself. For example, 8 squared is 64 (8 x 8).
7. Cube of Any Number
To find the cube of any number, multiply the number by itself twice. For example, 5 cubed is 125 (5 x 5 x 5).
8. Adding Fractions
When adding fractions with the same denominator, simply add the numerators and keep the denominator the same. For example, 3/5 + 2/5 = 5/5 = 1.
9. Subtracting Fractions
When subtracting fractions with the same denominator, simply subtract the numerators and keep the denominator the same. For example, 4/7 – 2/7 = 2/7.
10. Dividing Fractions
To divide fractions, multiply the first fraction by the reciprocal of the second fraction. For example, 2/3 ÷ 4/5 = 2/3 x 5/4 = 10/12 = 5/6.
11. Simplifying Fractions
To simplify a fraction, divide both the numerator and denominator by their greatest common divisor. For example, 12/36 can be simplified to 1/3 (divide both by 12).
12. Multiplying Decimals
To multiply decimals, ignore the decimal point and multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the original numbers and place the decimal point in the product accordingly. For example, 2.5 multiplied by 1.2 is 3 (25 x 12 = 300) with one decimal place.
13. Dividing Decimals
To divide decimals, move the decimal point in the divisor to the right until it becomes a whole number. Then, move the decimal point in the dividend the same number of places. Perform the division as if the decimal point wasn’t there. For example, 0.6 ÷ 0.3 is equal to 6 ÷ 3 = 2.
14. Multiplying Large Numbers
Break down large numbers into smaller parts and perform multiplication step by step. For example, to multiply 23 by 32, break it down into (20 + 3) multiplied by (30 + 2), which equals (20 x 30) + (20 x 2) + (3 x 30) + (3 x 2) = 600 + 40 + 90 + 6 = 736.
15. Squaring Two-Digit Numbers
When squaring two-digit numbers, multiply the tens digit by itself plus one, add a zero in the middle, and square the units digit. For example, 42 squared is (4 x 5) | 2 squared = 1764 (20 | 4 = 204, 4 squared = 16).
16. Finding Percentages
To find a percentage of a number, multiply the number by the percentage divided by 100. For example, 25% of 80 is (25/100) x 80 = 0.25 x 80 = 20.
17. Multiplying by 25
When multiplying a number by 25, divide the number by 4 and add two zeros at the end. For example, 36 multiplied by 25 is (36 ÷ 4) | 00 = 900 (9 | 00 = 900).
18. Multiplying by 50
When multiplying a number by 50, divide the number by 2 and add two zeros at the end. For example, 72 multiplied by 50 is (72 ÷ 2) | 00 = 3600 (36 | 00 = 3600).
19. Multiplying by 125
When multiplying a number by 125, divide the number by 8 and add three zeros at the end. For example, 32 multiplied by 125 is (32 ÷ 8) | 000 = 4000 (4 | 000 = 4000).
20. Squaring Numbers with 1 as the Tens Digit
When squaring numbers with 1 as the tens digit, multiply the units digit by itself, add the units digit at the end, and add a 1 in front. For example, 11 squared is 121 (1 squared = 1, add 1 at the end, and add 1 in front).
Tricks for Basic Operations
Addition Tricks
1. The Commutative Property
The commutative property of addition states that changing the order of the numbers being added does not change the sum. For example, 3 + 5 is the same as 5 + 3. This property can be useful when adding multiple numbers, as you can rearrange them to make the calculation easier.
2. Making Tens
When adding numbers, it can be helpful to make tens to simplify the calculation. For example, when adding 6 + 8, you can break down 8 into 2 + 6. Now, you have 6 + 2 (which is 8) + 6 (which is 12), resulting in a total of 14. This technique is particularly useful for mental calculations.
Subtraction Tricks
3. Borrowing from Neighbors
When subtracting numbers, if the digit in the column being subtracted is smaller than the corresponding digit in the minuend (the number being subtracted from), you can borrow from the neighbor (the digit to the left) to make the calculation easier.
For example, when subtracting 45 from 62, you can borrow 1 from the tens place, making it 52 – 45. This simplifies the calculation to 7 – 5, resulting in an answer of 7.
4. Subtracting from 9
Subtracting a number from 9 can be done by subtracting each digit from 9 and then subtracting the difference from 9. For example, when subtracting 7 from 9, you can subtract 7 from 9 to get 2, and then subtract 2 from 9 to get 7.
Multiplication Tricks
5. Multiplying by Powers of 10
Multiplying a number by a power of 10 is as simple as adding the appropriate number of zeros to the end of the original number.
For example, when multiplying 5 by 100, you can add two zeros to 5, resulting in 500.
6. The Distributive Property
The distributive property of multiplication states that multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the two numbers and then adding the products.
For example, 2 x (4 + 3) is the same as (2 x 4) + (2 x 3), which simplifies to 14.
Division Tricks
7. Estimating Quotients
Estimating the quotient can be helpful in quickly determining an approximate answer. For example, when dividing 87 by 5, you can estimate that the quotient will be around 17 (since 5 x 20 = 100). This gives you a rough idea of the answer without going through the entire division process.
8. Dividing by Powers of 10
Dividing a number by a power of 10 is as simple as moving the decimal point to the left by the appropriate number of places.
For example, when dividing 350 by 100, you can move the decimal point two places to the left, resulting in 3.5.
Other Basic Operation Tricks
9. Squaring Numbers Ending in 5
To square a number ending in 5, multiply the digit before 5 by its successor and append 25 at the end. For example, 35^2 = (3 x 4) + 25 = 1225.
10. Finding Percentages
When finding a percentage of a number, multiply the number by the percentage and divide by 100. For example, 20% of 80 is calculated as (20 x 80) / 100 = 16.
11. Checking Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 543 is divisible by 3 because 5 + 4 + 3 = 12, which is divisible by 3.
12. Checking Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 891 is divisible by 9 because 8 + 9 + 1 = 18, which is divisible by 9.
Tricks for Solving Equations
1. The Addition Property of Equality
The Addition Property of Equality states that if we add the same number to both sides of an equation, the equality is preserved.
For example, in the equation 2x + 5 = 11, we can subtract 5 from both sides to isolate the variable: 2x = 6.
Example:
Solve the equation 3x – 7 = 14.
By adding 7 to both sides, we get 3x = 21. Therefore, x = 7.
2. The Multiplication Property of Equality
The Multiplication Property of Equality states that if we multiply both sides of an equation by the same non-zero number, the equality is preserved.
For example, in the equation 4y = 16, we can divide both sides by 4 to find the value of y: y = 4.
Example:
Solve the equation 2(x – 3) = 10.
By dividing both sides by 2, we get x – 3 = 5. Adding 3 to both sides gives us x = 8.
3. The Subtraction Property of Equality
The Subtraction Property of Equality states that if we subtract the same number from both sides of an equation, the equality is preserved.
For example, in the equation 3z + 8 = 20, we can subtract 8 from both sides to isolate the variable: 3z = 12.
Example:
Solve the equation 5y + 2 = 17.
By subtracting 2 from both sides, we get 5y = 15. Therefore, y = 3.
4. The Division Property of Equality
The Division Property of Equality states that if we divide both sides of an equation by the same non-zero number, the equality is preserved.
For example, in the equation 2a = 10, we can divide both sides by 2 to find the value of a: a = 5.
Example:
Solve the equation 6(x + 4) = 42.
By dividing both sides by 6, we get x + 4 = 7. Subtracting 4 from both sides gives us x = 3.
5. The Distributive Property
The Distributive Property allows us to multiply a number by each term inside a set of parentheses. For example, in the equation 3(x + 2) = 15, we can use the distributive property to simplify the equation: 3x + 6 = 15.
Example:
Solve the equation 2(4x – 3) = 14.
By applying the distributive property, we get 8x – 6 = 14. Adding 6 to both sides gives us 8x = 20. Therefore, x = 2.5.
6. Combining Like Terms
When solving equations, it is important to combine like terms to simplify the equation. For example, in the equation 2x + 3x = 15, we can combine the x terms to get 5x = 15.
Example:
Solve the equation 4y – 2y + 5 = 17.
By combining like terms, we get 2y + 5 = 17. Subtracting 5 from both sides gives us 2y = 12. Therefore, y = 6.
7. Clearing Fractions
When dealing with equations that contain fractions, it can be helpful to clear the fractions by multiplying both sides of the equation by the least common denominator (LCD).
For example, in the equation 2/3x = 8, we can multiply both sides by 3 to eliminate the fraction: 2x = 24.
Example:
Solve the equation (1/2)x – 3 = 5.
By multiplying both sides by 2, we get x – 6 = 10. Adding 6 to both sides gives us x = 16.
8. Cross Multiplication
When dealing with equations that contain fractions, cross multiplication can be a useful technique. For example, in the equation 3/4x = 9/12, we can cross multiply to find the value of x: 4x = 27.
Example:
Solve the equation (2/3)x = 4/5.
By cross multiplying, we get 5x = 8. Therefore, x = 8/5.
9. Factoring
Factoring involves expressing an equation as a product of two or more expressions. For example, in the equation x^2 – 4x = 0, we can factor out an x to solve for x: x(x – 4) = 0.
Example:
Solve the equation x^2 + 5x + 6 = 0.
By factoring, we get (x + 2)(x + 3) = 0. Therefore, x = -2 or x = -3.
10. Quadratic Formula
The quadratic formula can be used to solve quadratic equations of the form ax^2 + bx + c = 0. For example, in the equation 2x^2 + 5x – 3 = 0, we can use the quadratic formula: x = (-b ± √(b^2 – 4ac)) / (2a).
Example:
Solve the equation x^2 – 4x + 4 = 0.
By applying the quadratic formula, we get x = (4 ± √(16 – 16)) / 2. Therefore, x = 2.
Tricks for Geometry
1. The Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it can be represented as:
a² + b² = c²
where ‘c’ represents the length of the hypotenuse, and ‘a’ and ‘b’ represent the lengths of the other two sides.
Example:
Consider a right-angled triangle with sides of length 3 units and 4 units. To find the length of the hypotenuse, we can use the Pythagorean theorem:
a = 3, b = 4
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = 5 units
2. The Angle Sum Property of Triangles
The sum of the interior angles of any triangle is always 180 degrees. This property can be useful in solving various problems related to angles in triangles.
Example:
Consider a triangle with interior angles measuring 60 degrees, 70 degrees, and ‘x’ degrees. To find the value of ‘x’, we can use the angle sum property:
60 + 70 + x = 180
130 + x = 180
x = 50 degrees
3. The Midpoint Formula
The midpoint formula allows us to find the coordinates of the midpoint of a line segment when given the coordinates of its endpoints. It can be expressed as:
[(x₁ + x₂) / 2, (y₁ + y₂) / 2]
Example:
Consider a line segment with endpoints A(2, 4) and B(6, 8). To find the coordinates of the midpoint, we can use the midpoint formula:
x₁ = 2, y₁ = 4, x₂ = 6, y₂ = 8
[(2 + 6) / 2, (4 + 8) / 2]
[8 / 2, 12 / 2]
[4, 6]
4. The Circle Area Formula
The area of a circle can be calculated using the formula:
πr²
where ‘r’ represents the radius of the circle.
Example:
Consider a circle with a radius of 5 units. To find the area of the circle, we can use the circle area formula:
r = 5
π(5)²
π(25)
25π
5. The Exterior Angle Sum Property of Polygons
The sum of the exterior angles of any polygon is always 360 degrees. This property can be helpful in finding the measure of each exterior angle of a polygon.
Example:
Consider a regular hexagon (a polygon with six equal sides and six equal angles). To find the measure of each exterior angle, we can use the exterior angle sum property:
Sum of exterior angles = 360 degrees
Each exterior angle = 360 degrees / Number of sides
Each exterior angle = 360 degrees / 6
Each exterior angle = 60 degrees
Tricks for Probability
1. Complement Rule
The complement rule states that the probability of an event not happening is equal to 1 minus the probability of the event happening.
For example, if the probability of rolling a 6 on a fair die is 1/6, then the probability of not rolling a 6 is 1 – 1/6 = 5/6.
Example:
Find the probability of drawing a card that is not a diamond from a standard deck of 52 playing cards. Since there are 13 diamonds in a deck, the probability of drawing a diamond is 13/52. Therefore, the probability of not drawing a diamond is 1 – 13/52 = 39/52.
2. Addition Rule
The addition rule states that the probability of either of two mutually exclusive events happening is equal to the sum of their individual probabilities.
Mutually exclusive events are events that cannot occur simultaneously. For example, the probability of rolling a 2 or a 4 on a fair die is 1/6 + 1/6 = 1/3.
Example:
Find the probability of drawing a heart or a diamond from a standard deck of 52 playing cards. There are 13 hearts and 13 diamonds in a deck, which are mutually exclusive. Therefore, the probability is 13/52 + 13/52 = 26/52 = 1/2.
3. Multiplication Rule
The multiplication rule states that the probability of two independent events happening is equal to the product of their individual probabilities.
Independent events are events that do not affect each other. For example, the probability of rolling a 2 on a fair die and flipping a head on a fair coin is 1/6 * 1/2 = 1/12.
Example:
Find the probability of drawing a spade from a standard deck of 52 playing cards and then drawing another spade without replacement.
The first probability is 13/52. After removing one spade, there are 12 spades left out of 51 cards. Therefore, the second probability is 12/51. So, the overall probability is 13/52 * 12/51 = 156/2652 = 1/17.
4. Conditional Probability
Conditional probability is the probability of an event happening given that another event has already occurred. It is calculated by dividing the probability of both events happening by the probability of the given event. For example, the probability of drawing a diamond given that the card drawn is red is 13/26.
Example:
Find the probability of drawing a king given that the card drawn is from the spades suit. There are 4 kings in a deck, and 13 spades. Therefore, the probability is 4/13.
5. Bayes’ Theorem
Bayes’ theorem is a formula used to calculate conditional probabilities. It provides a way to update probabilities based on new information.
The formula is: P(A|B) = (P(B|A) * P(A)) / P(B), where P(A|B) is the probability of event A happening given event B has occurred, P(B|A) is the probability of event B happening given event A has occurred, P(A) is the probability of event A happening, and P(B) is the probability of event B happening.
Example:
A medical test for a certain disease is 99% accurate. If a person has the disease, the test correctly identifies it 99% of the time. If a person does not have the disease, the test incorrectly identifies it 1% of the time. Suppose 1% of the population has the disease.
What is the probability that a person has the disease given that the test is positive? Let A be the event of having the disease, and B be the event of testing positive. P(A|B) = (P(B|A) * P(A)) / P(B) = (0.99 * 0.01) / ((0.99 * 0.01) + (0.01 * 0.99)) = 0.5. Therefore, the probability is 50%.
Tricks for Statistics
1. The Mean of a Set of Numbers
The mean, also known as the average, is a commonly used statistical measure. It is calculated by summing up all the numbers in a set and dividing the sum by the total count of numbers. For example, given the set {2, 4, 6}, the mean is calculated as (2 + 4 + 6) / 3 = 4.
2. The Median of a Set of Numbers
The median is the middle value in a set of numbers when arranged in ascending or descending order. If the set has an odd number of values, the median is the middle value itself.
For example, in the set {1, 3, 5, 7, 9}, the median is 5. If the set has an even number of values, the median is the average of the two middle values. For example, in the set {2, 4, 6, 8}, the median is (4 + 6) / 2 = 5.
3. The Mode of a Set of Numbers
The mode is the value that appears most frequently in a set of numbers. For example, in the set {2, 4, 6, 4, 8}, the mode is 4. A set can have multiple modes if multiple values appear with the same highest frequency.
4. The Range of a Set of Numbers
The range is the difference between the largest and smallest values in a set. For example, in the set {2, 4, 6, 8}, the range is 8 – 2 = 6.
5. The Variance of a Set of Numbers
The variance measures the spread of a set of numbers. It is calculated by taking the average of the squared differences between each value and the mean. For example, given the set {2, 4, 6}, the variance is calculated as [(2 – 4)^2 + (4 – 4)^2 + (6 – 4)^2] / 3 = 2.
6. The Standard Deviation of a Set of Numbers
The standard deviation is the square root of the variance. It provides a measure of how spread out the numbers in a set are. For example, given the variance of 2, the standard deviation is the square root of 2, which is approximately 1.41.
7. The Binomial Coefficient
The binomial coefficient, denoted as nCk, represents the number of ways to choose k items from a set of n items without regard to their order.
It is calculated using the formula n! / (k! * (n-k)!), where n! represents the factorial of n. For example, the binomial coefficient 4C2 is calculated as 4! / (2! * (4-2)!) = 6.
8. The Complement Rule
The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. For example, if the probability of raining tomorrow is 0.3, then the probability of not raining is 1 – 0.3 = 0.7.
9. The Addition Rule
The addition rule states that the probability of either of two mutually exclusive events occurring is equal to the sum of their individual probabilities.
For example, if the probability of rolling a 1 on a fair six-sided die is 1/6 and the probability of rolling a 2 is also 1/6, then the probability of rolling either a 1 or a 2 is 1/6 + 1/6 = 1/3.
10. The Multiplication Rule
The multiplication rule states that the probability of two independent events occurring together is equal to the product of their individual probabilities.
For example, if the probability of flipping a heads on a fair coin is 1/2 and the probability of rolling a 6 on a fair six-sided die is 1/6, then the probability of flipping a heads and rolling a 6 is 1/2 * 1/6 = 1/12.
11. The Factorial
The factorial of a positive integer n, denoted as n!, is the product of all positive integers from 1 to n. For example, 4! = 4 * 3 * 2 * 1 = 24.
12. The Permutation
A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken k at a time is denoted as nPk and is calculated as n! / (n-k)!.
For example, the number of permutations of 4 objects taken 2 at a time is 4! / (4-2)! = 12.
13. The Combination
A combination is a selection of objects without regard to their order. The number of combinations of n objects taken k at a time is denoted as nCk and is calculated as n! / (k! * (n-k)!).
For example, the number of combinations of 4 objects taken 2 at a time is 4! / (2! * (4-2)!) = 6.
14. The Law of Large Numbers
The law of large numbers states that as the number of trials or observations increases, the observed probability of an event approaches its theoretical probability. This law forms the basis for statistical inference and estimation.
15. The Central Limit Theorem
The central limit theorem states that the distribution of the sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
This theorem is fundamental in hypothesis testing and confidence interval estimation.
16. The Z-Score
The z-score measures the number of standard deviations a data point is away from the mean. It is calculated as (x – μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation.
A z-score of 0 indicates that the data point is equal to the mean, while a z-score of -1 or +1 indicates that the data point is one standard deviation below or above the mean, respectively.
17. The T-Distribution
The t-distribution is a probability distribution that is used when the sample size is small or the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails.
The t-distribution is commonly used in hypothesis testing and confidence interval estimation.
18. The F-Distribution
The F-distribution is a probability distribution that arises in the analysis of variance (ANOVA) and regression. It is used to compare the variances of two or more groups.
The F-distribution is positively skewed and has a shape that depends on the degrees of freedom.
19. The Chi-Square Test
The chi-square test is a statistical test that is used to determine whether there is a significant association between two categorical variables.
It compares the observed frequencies with the expected frequencies under the assumption of independence. The chi-square test is commonly used in goodness-of-fit tests and tests of independence.
20. The Correlation Coefficient
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative correlation, +1 indicates a perfect positive correlation, and 0 indicates no correlation. The correlation coefficient is commonly used in regression analysis and correlation studies.
Tricks for Trigonometry
1. Pythagorean Identity
The Pythagorean Identity is a fundamental equation in trigonometry that relates the three basic trigonometric functions: sine, cosine, and tangent. It states that for any angle θ, sin²(θ) + cos²(θ) = 1.
Example:
For an angle of 45 degrees, sin²(45) + cos²(45) = 1. This equation holds true for all angles.2. Unit Circle
The unit circle is a powerful tool in trigonometry that allows you to easily find the values of trigonometric functions for any angle. By memorizing the coordinates of points on the unit circle, you can quickly determine the values of sine, cosine, and tangent.
Example:
For an angle of 30 degrees, the coordinates on the unit circle are (√3/2, 1/2). Therefore, sin(30) = 1/2 and cos(30) = √3/2.3. Radian to Degree Conversion
To convert an angle from radians to degrees, multiply it by 180/π. This trick comes in handy when working with trigonometric functions that require angles in degrees.
Example:
An angle of π/4 radians is equivalent to 45 degrees.4. Degree to Radian Conversion
To convert an angle from degrees to radians, multiply it by π/180. This trick is useful when working with trigonometric functions that require angles in radians.
Example:
An angle of 60 degrees is equivalent to π/3 radians.5. Trig Identities
Trig identities are equations involving trigonometric functions that are true for all values of the variables. Memorizing these identities can help simplify complex trigonometric expressions.
Example:
The double angle identity for sine states that sin(2θ) = 2sin(θ)cos(θ).6. Trig Ratios of Complementary Angles
The trigonometric ratios of complementary angles (angles that add up to 90 degrees) are equal. This trick allows you to find the value of one trigonometric function if you know the value of its complementary function.
Example:
If sin(θ) = 1/2, then cos(θ) = 1/2.7. Sum and Difference Formulas
The sum and difference formulas allow you to express the trigonometric functions of the sum or difference of two angles in terms of the trigonometric functions of the individual angles.
Example:
The sum formula for sine states that sin(α + β) = sin(α)cos(β) + cos(α)sin(β).8. Periodicity of Trigonometric Functions
Trigonometric functions are periodic, meaning they repeat their values after a certain interval. Understanding the periodicity of trigonometric functions can help simplify calculations.
Example:
The sine function is periodic with a period of 2π. Therefore, sin(θ + 2π) = sin(θ).9. Trig Functions of Negative Angles
The trigonometric functions of negative angles are equal to the trigonometric functions of their positive counterparts.
Example:
sin(-θ) = -sin(θ)10. Trig Functions of π/2 ± θ
The trigonometric functions of angles in the form of π/2 ± θ can be easily determined by swapping the values of sine and cosine.
Example:
sin(π/2 - θ) = cos(θ)11. Trig Functions of 2π – θ
The trigonometric functions of angles in the form of 2π – θ are equal to the negative trigonometric functions of θ.
Example:
cos(2π - θ) = -cos(θ)12. Trig Functions of π/2 + θ
The trigonometric functions of angles in the form of π/2 + θ can be easily determined by swapping the values of sine and cosine.
Example:
tan(π/2 + θ) = cot(θ)13. Trig Functions of π – θ
The trigonometric functions of angles in the form of π – θ are equal to the negative trigonometric functions of θ.
Example:
sin(π - θ) = -sin(θ)14. Trig Functions of 3π/2 + θ
The trigonometric functions of angles in the form of 3π/2 + θ can be easily determined by swapping the values of sine and cosine.
Example:
cos(3π/2 + θ) = -sin(θ)15. Trig Functions of 2π + θ
The trigonometric functions of angles in the form of 2π + θ are equal to the trigonometric functions of θ.
Example:
tan(2π + θ) = tan(θ)16. Trig Functions of 3π/2 – θ
The trigonometric functions of angles in the form of 3π/2 – θ can be easily determined by swapping the values of sine and cosine.
Example:
sin(3π/2 - θ) = -cos(θ)17. Trig Functions of -θ
The trigonometric functions of -θ are equal to the negative trigonometric functions of θ.
Example:
cos(-θ) = cos(θ)18. Trig Functions of 2πn ± θ
The trigonometric functions of angles in the form of 2πn ± θ are equal to the trigonometric functions of θ, where n is an integer.
Example:
sin(2πn + θ) = sin(θ)19. Trig Functions of -θ ± 2πn
The trigonometric functions of angles in the form of -θ ± 2πn are equal to the trigonometric functions of θ, where n is an integer.
Example:
tan(-θ + 2πn) = tan(θ)20. Trig Functions of -π/2 ± θ
The trigonometric functions of angles in the form of -π/2 ± θ can be easily determined by swapping the values of sine and cosine.
Example:
cos(-π/2 + θ) = sin(θ)Conclusion
These 100+ mathematical tricks are just a glimpse into the vast world of mathematics. By mastering these tricks, high school learners can build a strong foundation for further studies in mathematics and related fields.
Remember, practice makes perfect, so keep honing your skills and exploring new mathematical concepts. With dedication and perseverance, you can conquer any mathematical challenge that comes your way!
