Question

How do you simplify \[\cos \left( {\dfrac{\pi }{2} - \theta } \right)\]?

Answer 1

Hint: To simplify the given trigonometric equation, we need to apply the Addition formula of trigonometric function, as it is of the form \[\cos \left( {A \pm B} \right)\], hence applying the formula as \[\cos \left( {A \pm B} \right) = \cos A\cos B \mp \sin A\sin B\], here we need to substitute the value of A and B, from the given equation, then using trigonometric identity we need to simplify and find the value of the given equation.

Formula used:\[\cos \left( {A \pm B} \right) = \cos A\cos B \mp \sin A\sin B\]

Complete step-by-step solution:
Given,
\[\cos \left( {\dfrac{\pi }{2} - \theta } \right)\]
This is a well-used trigonometric relation along with\[\sin \left( {\dfrac{\pi }{2} - \theta } \right)\] i.e.,
\[\cos \left( {\dfrac{\pi }{2} - \theta } \right) = \sin \theta \] and \[\sin \left( {\dfrac{\pi }{2} - \theta } \right) = \cos \theta \].
This implies that: sin(angle) = cos(complement) and cos(angle) = sin(complement)
However, we need to simplify the above question using the appropriate Addition formula as:
\[\cos \left( {A \pm B} \right) = \cos A\cos B \mp \sin A\sin B\] ………………. 1
Hence, now substitute the value of A and B as, \[A = \dfrac{\pi }{2}\]and \[B = \theta \] in equation 1, as per the given equation, we get:
\[ \Rightarrow \cos \left( {\dfrac{\pi }{2} - \theta } \right) = \cos \left( {\dfrac{\pi }{2}} \right)\cos \theta + \sin \left( {\dfrac{\pi }{2}} \right)\sin \theta \] ………………………. 2
We know that, \[\cos \left( {\dfrac{\pi }{2}} \right) = 0\] and \[\sin \left( {\dfrac{\pi }{2}} \right) = 1\], hence substituting it in equation 2 we get:
\[ \Rightarrow \cos \left( {\dfrac{\pi }{2} - \theta } \right) = 0 \times \cos \theta + 1 \times \sin \theta \]
Simplifying the terms, we get:
\[ \Rightarrow \cos \left( {\dfrac{\pi }{2} - \theta } \right) = \sin \theta \]

Additional information: In trigonometry sin, cos and tan values are the primary functions we consider while solving trigonometric problems. These trigonometry values are used to measure the angles and sides of a right-angle triangle. Apart from sine, cosine and tangent values, other values are cotangent, secant and cosecant.

Note: The key point to find the values of any trigonometric equation is to note all the formulas related to all the trigonometric identity and calculate all the terms asked. And here are some of the formulas to be noted if the equation consists of sin, cos and tan and we are asked to simplify then we have:
\[\cos \left( {A \pm B} \right) = \cos A\cos B \mp \sin A\sin B\]
\[\sin \left( {A \pm B} \right) = \sin A\cos B \pm \cos A\sin B\]
\[\tan \left( {A \pm B} \right) = \dfrac{{\tan A \pm \tan B}}{{1 \mp \tan A\tan B}}\]