Question

How do you find \[\sin {{65}^{0}}\cdot \cos {{38}^{0}}\]?

Answer 1

Hint: In the given question, we have been asked to find the value of the given trigonometric expression. In order to find the value of a given expression, first we need to simplify the given expression so that we can apply the trigonometric identity. After rewritten, apply the trigonometric identity that is \[2\sin a\cos b=\sin \left( a+b \right)+\sin \left( a-b \right)\]. Later substituting the values, solve the given expression and then find the value of a function by using the calculator. In this way we will get the required answer.

Complete step by step solution:
We have given that,
\[\sin {{65}^{0}}\cdot \cos {{38}^{0}}\]
Rewrite the above expression as,
\[\dfrac{1}{2}\left( 2\sin {{65}^{0}}\cdot \cos {{38}^{0}} \right)\]
Using the trigonometric identity of multiplication of sine and cosine function i.e.
\[2\sin a\cos b=\sin \left( a+b \right)+\sin \left( a-b \right)\]
Applying the identity in the given expression, we obtained,
\[\dfrac{1}{2}\left( 2\sin {{65}^{0}}\cdot \cos {{38}^{0}} \right)=\dfrac{1}{2}\left( \sin \left( {{65}^{0}}+{{38}^{0}} \right)+\sin \left( {{65}^{0}}-{{38}^{0}} \right) \right)\]
Solving the above expression, we get
\[\dfrac{1}{2}\left( 2\sin {{65}^{0}}\cdot \cos {{38}^{0}} \right)=\dfrac{1}{2}\left( \sin \left( {{103}^{0}} \right)+\sin \left( {{27}^{0}} \right) \right)\]
Solving the above expression, we substituting the values of the given trigonometric function by using the calculator, we obtained
\[\sin {{65}^{0}}\cdot \cos {{38}^{0}}=0.7142\]

Therefore, The value of \[\sin {{65}^{0}}\cdot \cos {{38}^{0}}\] is equal to \[0.7142\].

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometry and the trigonometric identities as well. It will make questions easier to solve and you will get the right answer. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.