
How do you find an expression sin(x) in terms of and ?
Answer
478.5k+ views
Hint:
In this question, we were asked to find the expression of sin x in terms of and . So, here we will use the McLaurin formula from the series of the exponential function. We will also use the basic properties of trigonometry. So let us see how we can solve this problem.
Complete step by step solution:
For solving the above question, we will use the McLaurin formula which is
So,
Let us suppose that n = 2k in the first case and n = 2k + 1 in the second case, so that we can separate the odd and even terms for n
Now see,
So,
We know that the McLaurin expansions of cos x and sin x, and thus we get
Which is Euler’s formula
Now, let us suppose that cos x is an even function and sin x is an odd function then we will get,
Finally, we will get
Therefore, on solving sin x in terms of and we get
Note:
There is an alternative way, to solve this problem, let us see that as well.
We know that (from Euler’s formula)
Also,
Since, cos(-x) = cos x and sin(-x) = -sin x
Now, we have
On adding both of them we get,
Now,
On subtracting both of them we get,
Then,
Therefore, even with this alternative way, we get the same solution that is
In this question, we were asked to find the expression of sin x in terms of
Complete step by step solution:
For solving the above question, we will use the McLaurin formula which is
So,
Let us suppose that n = 2k in the first case and n = 2k + 1 in the second case, so that we can separate the odd and even terms for n
Now see,
So,
We know that the McLaurin expansions of cos x and sin x, and thus we get
Which is Euler’s formula
Now, let us suppose that cos x is an even function and sin x is an odd function then we will get,
Finally, we will get
Therefore, on solving sin x in terms of
Note:
There is an alternative way, to solve this problem, let us see that as well.
We know that
Also,
Since, cos(-x) = cos x and sin(-x) = -sin x
Now, we have
On adding both of them we get,
Now,
On subtracting both of them we get,
Then,
Therefore, even with this alternative way, we get the same solution that is
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