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Proof of angle addition formula for sine | Trigonometry
Trigonometric Addition Formulas
Angle addition formulas express trigonometric functions of sums of angles
The fundamental formulas of angle addition in trigonometry are given by
The first four of these are known as the prosthaphaeresis formulas, or sometimes as Simpson’s formulas.
The sine and cosine angle addition identities can be compactly summarized by the matrix equation
These formulas can be simply derived using complex exponentials and the Euler formula as follows.
Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting
Taking the ratio of (1) and (3) gives the tangent angle addition formula
The double-angle formulas are
Multiple-angle formulas are given by
and can also be written using the recurrence relations
The angle addition formulas can also be derived purely algebraically without the use of complex numbers. Consider the small right triangle in the figure above, which gives
Now, the usual trigonometric definitions applied to the large right triangle give
Solving these two equations simultaneously for the variables
These can be put into the familiar forms with the aid of the trigonometric identities
and
which can be verified by direct multiplication. Plugging (◇) into (◇) and (38) into (◇) then gives
as before.
A similar proof due to Smiley and Smiley uses the left figure above to obtain
Similar diagrams can be used to prove the angle subtraction formulas (Smiley 1999, Smiley and Smiley). In the figure at left,
A more complex diagram can be used to obtain a proof from the
An interesting identity relating the sum and difference tangent formulas is given by
Solved Examples
Example 1: Find the sum of 67 and 15.
Solution:
Given :
First number = 67, second number = 15
Sum = First number + Second number
Example 2: Find the sum of 413, 78 and 350.
Solution:
Explanation:
The digits at units place are 3, 8, 0
3 + 8 + 0 = 11
Hence, keep1 and carry 1 for the addition of digits at tens place
Sum of digits at tens place = 1 + 7 + 5 + 1(carried) = 14
Here, keep 4 and carry 1 for the next addition of digits at hundreds place
Sum of digits at hundreds place = 4 + 3 + 1 (carried) = 8
Therefore, the result is 841.
What is Addition in Maths?
Addition is an operation used in math to add numbers. The result that is obtained after addition is known as the sum of the given numbers. For example, if we add 2 and 3, (2 + 3) we get the sum as 5. Here, we performed the addition operation on two numbers 2 and 3 to get the sum, i.e., 5
Addition Definition
Addition is defined as the process of calculating the total of two or more numbers. This calculation can be a simple one or a process that involves regrouping and carrying over of numbers.
Addition Symbol
In mathematics, we have different symbols. The addition symbol is one of the widely used math symbols. In the above definition of addition, we read about adding two numbers 2 and 3. If we observe the pattern of addition (2 + 3 = 5) the symbol (+) connects the two numbers and completes the given expression. The addition symbol consists of one horizontal line and one vertical line. It is also known as the addition sign or the plus sign (+)
Parts of Addition
An addition statement can be split into the following parts.
- Addend: The numbers that are added are known as the addends.
- Addition Symbol: There is the addition symbol (+) which is placed in between the addends. If the statement is written horizontally as shown below, then we place an equal to sign (=) just before the sum is written.
- Sum: The final result obtained after adding the addends is known as the sum.
How to Solve Addition Sums?
While solving addition sums, one-digit numbers can be added in a simple way, but for larger numbers, we split the numbers into columns using their respective place values, like ones, tens, hundreds, thousands, and so on. We always start doing addition from the right side as per the place value system. This means we start from the ones column, then move on to the tens column, then to the hundreds column, and so on. While solving such problems we may come across some cases with carry-overs and some without carry-overs. Let us understand addition with regrouping and addition without regrouping in the following sections.
Addition Without Regrouping
The addition in which the sum of the digits is less than or equal to 9 in each column is called addition without regrouping. Let us understand how to add two or more numbers without regrouping with the help of an example.
Example: Add 11234 and 21123
Solution: We will use the following given steps and try to relate them with the following figure.
- Step 1: Start with the digits in ones column. (4 + 3 = 7)
- Step 2: Move to the digits in tens column. (3 + 2 = 5)
- Step 3: Now add the digits in hundreds column. (2 + 1 = 3)
- Step 4: After this, add the digits in thousands column. (1 + 1 = 2)
- Step 5: Finally, add the digits in ten thousands column. (1 + 2 = 3)
- Step 6: 11234 + 21123 = 32357
In addition without regrouping, we simply add the digits in each place value column and combine the respective sums together to get the answer. Now, let us understand addition with regrouping.
Addition With Regrouping
While adding numbers, if the sum of the addends is greater than 9 in any of the columns, we regroup this sum into tens and ones. Then we carry over the tens digit of the sum to the preceding column and write the ones digit of the sum in that particular column. In other words, we write only the number in ‘ones place digit’ in that particular column, while taking the ‘tens place digit’ to the column to the immediate left. Let us understand how to add two or more numbers by regrouping with the help of an example.
Example: Add 3475 and 2865.
Solution: Let us follow the given steps and try to relate them with the following figure.
- Step 1: Start with the digits in ones place. (5 + 5 = 10). Here the sum is 10. The tens digit of the sum, that is, 1, will be carried to the preceding column.
- Step 2: Add the digits in the tens column along with the carryover 1. This means, 1 (carry-over) + 7 + 6 = 14. Here the sum is 14. The tens digit of the sum, that is, 1, will be carried to the hundreds column.
- Step 3: Now, add the digits in the hundreds place along with the carryover digit 1. This means, 1 (carry-over) + 4 + 8 = 13. Here the sum is 13. The tens digit of the sum, that is, 1, will be carried to the thousands column.
- Step 4: Now, add the digits in the thousands place along with the carryover digit 1, that is, 1 (carry-over) + 3 + 2 = 6
- Step 5: Therefore, the sum of 3475 + 2865 = 6340
Note: There is an important property of addition which states that changing the order of numbers does not change the answer. For example, if we reverse the addends of the above illustration we will get the same sum as a result (2865 + 3475 = 6340). This is known as the commutative property of addition.
Number Line Addition
Another way to add numbers is with the help of number lines. Let us understand the addition on a number line with the help of an example and the number line given below.
Example: Add 10 + 3 using a number line
Solution: We start by marking the number 10 on the number line. When we add using a number line, we count by moving one number at a time to the right of the number. Since we are adding 10 and 3, we will move 3 steps to the right. This brings us to 13. Hence, 10 + 3 = 13.
Addition Properties
While performing addition we commonly use the properties listed below:
- Commutative Property: According to this property, the sum of two or more addends remains the same irrespective of the order of the addends. For example, 8 + 7 = 7 + 8 = 15
- Associative Property: According to this property, the sum of three or more addends remains the same irrespective of the grouping of the addends. For example, 5 + (7 + 3) = (5 + 7) + 3 = 15
- Additive Identity Property: According to this property of addition, if we add 0 to any number, the resultant sum is always the actual number. For example, 0 + 7 = 7.
Addition Word Problems
The concept of the addition operation is used in our day-to-day activities. We should carefully observe the situation and identify the solution using the tips and tricks that follows addition. Let us understand how to solve addition word problems with the help of an interesting example.
Example: A soccer match had 4535 spectators in the first row and 2332 spectators in the second row. Using the concept of addition find the total number of spectators present in the match.
Solution:
The number of spectators in the first row = 4535; the number of spectators in the second row = 2332. We can get the total number of spectators if we add the given number of spectators in the two rows.
Here 4535 and 2332 are the addends. Let us find the total number of spectators by adding these two numbers using the following steps.
- Step 1: Add the digits in the ones place. (5 + 2 = 7)
- Step 2: Add the digits in the tens place. (3 + 3 = 6)
- Step 3: Add the digits in the hundreds place. (5 + 3 = 8)
- Step 4: Now add digits in the thousands place. (4 + 2 = 6)
- Step 5: 4535 + 2332 = 6867
Therefore, the total number of spectators present in the match = 6867
Here are a few tips and tricks that you can follow while performing addition in your everyday life.
Tips and Tricks on Addition
- Words like ‘put together, ‘in all’, ‘altogether’, ‘total’ give a clue that you need to add the given numbers.
- Start with the larger number and add the smaller number to it. For example, adding 12 to 43 is easier than adding 43 to 12.
- Break numbers according to their place values to make addition easier. For example, 22 + 64 can be split as 20 + 2 + 60 + 4. While this looks difficult, it makes mental addition easier.
- When adding different digit numbers, make sure to place the numbers one below the other in the correct column of their place value.
- Adding zero to any number gives the number itself.
- When 1 is added to any number, the sum is the successor of that number.
- The sign used to denote addition is ‘+’
- The order in which you add a set of numbers doesn’t matter, the sum remains the same. For example, 2 + 5 + 3 = 10; and 5 + 3 + 2 = 10. It is called the associative property of addition.
Addition Examples
Example 1: 8 bees set off to suck nectar from the flowers. Soon 7 more joined them. Use addition to find the total number of bees who went together to suck nectar.
Solution:Number of bees who set off to suck nectar = 8Number of bees who joined them = 7Therefore, on performing addition, the total number of bees who went together were: 8 + 7= 15.
Answer: 15 bees
Example 2: Using addition tricks, solve the following addition word problem.Jerry collected 89 seashells and Eva collected 54 shells. How many seashells did they collect in all?Solution:Number of shells collected by Jerry = 89Number of shells collected by Eva = 54Therefore, the total number of sea shells collected by both of them = 89 + 54 = 143
Answer: 143 seashells
Example 3: During an annual Easter egg hunt, the participants found 2403 eggs in the clubhouse, 50 easter eggs in the park, and 12 easter eggs in the town hall. Can you find out how many eggs were found in that day’s hunt using the concept of addition?
Solution:
Number of easter eggs found in the Clubhouse = 2403
Number of easter eggs found in the park = 50
Number of easter eggs found in the Town Hall = 12
We write the numbers into columns according to their place values of ones, tens, hundreds, thousands and then add them:
Answer: Therefore, the total number of eggs found in that day’s hunt is 2465.
FAQs on Addition
What is Addition in Math?
Addition is the process of adding two or more numbers together to get their sum. Addition in math is a primary arithmetic operation, used for calculating the total of two or more numbers. For example, 7 + 6 = 13.
Where do we use Addition?
We use addition in our everyday situations. For example, if we want to know how much money we spent on the items we bought, or we want to calculate the time we would take to finish a task, or we want to know the number of ingredients used in cooking something, we need to perform the addition operation.
What are the Types of Addition?
The types of addition mean the various methods used in addition. For example, vertical addition, addition using number charts, the addition of small numbers using your fingers, addition using number line, and so on.
What are Addition Strategies?
Addition strategies are the different ways in which addition can be learned. For example, using a number line, with the help of a place value chart, separating the tens and ones and then adding them separately, and many others.
What are the Real-Life Examples of Addition?
There are many addition examples that we come across in our day-to-day lives. Suppose you have 5 apples, and your friend gave you 3 more, after adding 5 + 3, we get 8. So, you have 8 apples altogether. Similarly, suppose there are 16 girls and 13 boys in a class, if we add the numbers 16 + 13, we get the total number of students in the class, which is 29.
What are the Properties of Addition?
The basic properties of addition are given below. Each property has its individual significance based on addition.
- Commutative Property: According to this property, the sum of two or more addends remains the same even if the order of the addends changes. For example, 3 + 7 = 7 + 3 = 10
- Associative Property: According to this property, the sum of three or more addends remains the same irrespective of the grouping of the addends. For example, (8 + 7) + 2 = 8 + (7 + 2) = 17
- Additive Identity Property: According to this property of addition, if we add 0 to any number, the resultant sum is always the actual number. For example, 0 + 16 = 16.
What are the Parts of Addition?
The different parts of addition are given below. Let us understand these parts with the help of an example. For example, let us take 4 + 7 + 2 = 13
- Addend: In addition, the numbers or terms that are added together are known as the addends. In this case, 4, 7, and 2 are the addends.
- Addition symbol (+) and the equal-to sign (=): The addition symbol is used in between the addends and the equal-to sign is placed just before the sum.
- Sum: The final result obtained after performing addition is known as the sum. Here, the sum is 13.
What is the Identity Property of Addition?
According to the identity property of addition, if 0 is added to any number, the resultant sum is always the actual number. For example, 0 + 16 = 16.
What is the Difference Between Addition and Subtraction?
Addition is a math operation in which we add the numbers together to get their sum. It is denoted by the addition symbol (+). For example, on adding 5 and 7 we get 12. This is represented as, 5 + 7 = 12. Subtraction is the arithmetic operation of calculating the difference between two numbers. It is denoted by the subtraction symbol (-). For example, if we subtract 8 from 19, we get 11. This is represented as 19 – 8 = 11.
How to Write an Addition Sentence?
An addition sentence is the way in which addition is expressed as a mathematical expression. An addition sentence consists of 2 or more values that are written together with the addition symbol (+) in between them and an equal sign (=) at the end just before the sum. For example, if we want to add 2, 4, and 5 we write the addition sentence as, 2 + 4 + 5 = 11, where, 2, 4 and 5 are called the addends and 11 is the sum.
How to do Addition with Regrouping?
Addition with regrouping happens when we carry over the extra digit to the next column. When we add numbers, and we get a sum which is greater than 9 in any of the columns, we regroup this sum into tens and ones. Then we carry over the tens digit of the sum to the preceding column and write the ones digit of the sum in that particular column. In other words, we write only the number in ‘ones place digit’ in that particular column, while we take the ‘tens place digit’ to the column to the immediate left. A detailed example of addition with regrouping is given above on this page.
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