
Find the value of
Answer
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Hint: We solve this problem first by converting all the angles given to less than because we have the trigonometric table of standard values for the angles less than
We convert each angle as the sum or difference of that we convert each angle as or so that we have conversions of trigonometric ratios as
By using the above formulas we reduce the given angles into angles less than to find the required value easily.
Complete step by step answer:
We are asked to find the value of
Let us assume that the required value as
Now, let us convert each angle in the above equation into or
Now, by converting the first angle that is we get
Now, by converting the next angle that is we get
Now, by converting the next angle that is we get
Now, by substituting the required angles in the equation (i) we get
We know that the conversions of trigonometric ratios that are
Now, by using these conversions in the above equation we get
We know that from the standard trigonometric table the values of trigonometric ratios for some angles as
Now, by substituting these values in above equation we get
Therefore we can conclude that the value of given expression as
Note: Students may do mistake in converting the given angles into angles less than
Here, we can see that the given angles are all near to so that we can easily convert the given angles in the form or to get the angles less than
But we have so many ways of converting the given angles into angles less than
We can convert in the form or
But in this process the formulas of conversions will change, that is sine gets converted to cosine and cosine gets converted to sine and also the sign changes.
But even though we will not get the all angles less than because
Here, we again need to convert the angle in the form of which increases the solution length.
So, we need to check for nearest value where we get the given angles as the angles less than
We convert each angle as the sum or difference of
By using the above formulas we reduce the given angles into angles less than
Complete step by step answer:
We are asked to find the value of
Let us assume that the required value as
Now, let us convert each angle in the above equation into
Now, by converting the first angle that is
Now, by converting the next angle that is
Now, by converting the next angle that is
Now, by substituting the required angles in the equation (i) we get
We know that the conversions of trigonometric ratios that are
Now, by using these conversions in the above equation we get
We know that from the standard trigonometric table the values of trigonometric ratios for some angles as
Now, by substituting these values in above equation we get
Therefore we can conclude that the value of given expression as
Note: Students may do mistake in converting the given angles into angles less than
Here, we can see that the given angles are all near to
But we have so many ways of converting the given angles into angles less than
We can convert in the form
But in this process the formulas of conversions will change, that is sine gets converted to cosine and cosine gets converted to sine and also the sign changes.
But even though we will not get the all angles less than
Here, we again need to convert the angle
So, we need to check for nearest value where we get the given angles as the angles less than
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