Trigonometric Addition and Subtraction formulae
Trigonometric Addition ( Sum ) and Subtraction ( Difference ) formula:
which are popularly
known as addition ( sum )
and subtraction( difference )
formulae are as follows:
Sine of sum of angles:
Cosine of sum of angles:
Sine of difference of angles:
Cosine of difference of angles:
And similarly the sum and difference of angle formula of Tangent are:
Proof of Trigonometric Sum and Difference Formulae:
Now let us prove the identities or formulae listed above.
In the approach to prove these identities we should first prove the identity of and rest of the identities can be derived from the identity:
Let and be two points , different from the origin , on the terminal arms of two angles A and B such that placed in the standard position as shown in the figure below:
If: and then,
Hence , Using the Distance formula:
or, ————– Expression 1 for PQ^2
Now , Let us rotate the coordinate system so that OQ coincides with the positive x-axis. This is equivalent to place the angle (A-B) in the standard position with OQ along the positive x-axis. As shown in figure below:
Then , the new coordinates of and are given by:
Or , ————– Expression 2 for PQ^2
Now comparing the two expression for , we have:
This is the subtraction or difference formula for cosine , true for arbitrary angles A and B.
Now we can derive other sum and difference formula using this formula, before deriving those formulae let us consider following special cases:
Note: All the numbers and variables eg: A , B , 90 are in Degrees , but we can also apply this in other angle measurement system by replacing the value of numbers used with corresponding value.
Case 1> If A=o then , we get:
Case 2> If A=90 then, we get:
Case 3> If B is replaced by 90-B in Case 2 then we get:
Case 4> If B is replaced by -90 and A is replaced by -A then , we get:
Now let us Derive other general formulas using the special cases above and cosine difference of angle formula:
Replacing B by -B in we get:
Now using case 2 and 5 we get:
Replacing A by 90-A to get:
Then , using Case 3 and 4 we get:
Replacing B by -B in above formula or and the uisng case 2 and 5 we get:
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