Question

Find the value of \[\sin \left( \alpha -\beta \right)\], If \[\cos \left( \alpha +\beta \right)=0\].
(a) \[\cos \beta \]
(b) \[\cos 2\beta \]
(c) \[\sin \alpha \]
(d) \[\sin 2\alpha \]

Answer 1

- Hint:- In this question, from the given condition on cosine we can get the sum of the angles and then the relation between the angles. Then by substituting this relation in the sine term mentioned we can get the result.





Complete step-by-step solution -

Given in the question that,
\[\Rightarrow \cos \left( \alpha +\beta \right)=0\]
From the trigonometric ratios of the compound angles we can say that the algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angles.
As we already know the cosine values for some of the standard angles we have,
\[\cos {{90}^{\circ }}=0\]
Now, from the given condition of the cosine term in the question we get,
\[\Rightarrow \cos \left( \alpha +\beta \right)=0\]
Now, on comparing this with the standard value of the cosine term we get,
\[\Rightarrow \alpha +\beta ={{90}^{\circ }}\]
Now, by rearranging the terms we can write one angle in terms of the other.
\[\Rightarrow \alpha ={{90}^{\circ }}-\beta \]
From the given question we need to find the value of the function.
\[\sin \left( \alpha -\beta \right)\]
Now, by substituting the value of \[\alpha \]above in terms of \[\beta \]in the above equation we get,
\[\begin{align}
  & \Rightarrow \sin \left( \alpha -\beta \right) \\
 & \Rightarrow \sin \left( {{90}^{\circ }}-\beta -\beta \right) \\
\end{align}\]
Now, on simplifying further and rewriting the above equation we get,
\[\Rightarrow \sin \left( {{90}^{\circ }}-2\beta \right)\]
As we already know that from the trigonometric ratios of allied angles
\[\sin \left( {{90}^{\circ }}-\theta \right)=\cos \theta \]
Now, by using the condition of trigonometric ratios of allied angles in the above equation obtained we get,
\[\Rightarrow \cos 2\beta \]
Hence, the correct option is (b).

Note: Instead of directly getting the relation between the angles by using the value of cosine for standard angles we can also expand it by using the trigonometric ratios of compound angles formula and then get the relation between the sine and the cosine terms.
Then again expand the sine term to be found by using the trigonometric ratios of compound angles formulae and then substitute the relation obtained by solving the first case. Both the methods give the same result but this method is lengthy and includes a lot of solving and conversions.