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Question: What is the value of \(\sin 30^\circ .\cos 60^\circ + \cos 30^\circ .\sin 60^\circ \)?...

What is the value of sin⁡30∘.cos⁡60∘+cos⁡30∘.sin⁡60∘\sin 30^\circ .\cos 60^\circ + \cos 30^\circ .\sin 60^\circ ?

Explanation

Solution

In this question, we have been asked to find the value of a trigonometric equation. To answer the question, you must be aware about the trigonometric table. Write down the table and simply put the values. Then, simplify and you will get the required answer.
You can use the other method also. In this method, think about the trigonometric formulae. You will notice that the given expression is just an expansion of the formula sin⁡(a+b)\sin \left( {a + b} \right). Recognize the values of aa and bb. Then, put them in the identity and you will have your answer directly. (See for the solution using this method in note below.)

Complete step-by-step solution:
We have been given a trigonometric equation and we have to find its value. First, we will draw the trigonometric table.

Ratio/ Angles0∘0^\circ 30∘30^\circ 45∘45^\circ 60∘60^\circ 90∘90^\circ
Sin0012\dfrac{1}{2}12\dfrac{1}{{\sqrt 2 }}32\dfrac{{\sqrt 3 }}{2}11
Cos1132\dfrac{{\sqrt 3 }}{2}12\dfrac{1}{{\sqrt 2 }}12\dfrac{1}{2}00

Now, we have all the required values handy. Next step is to simply put the values in the equation.
⇒sin⁡30∘.cos⁡60∘+cos⁡30∘.sin⁡60∘\Rightarrow \sin 30^\circ .\cos 60^\circ + \cos 30^\circ .\sin 60^\circ
Putting all the values,
⇒12×12+32×32\Rightarrow \dfrac{1}{2} \times \dfrac{1}{2} + \dfrac{{\sqrt 3 }}{2} \times \dfrac{{\sqrt 3 }}{2}
Simplifying the equation,
⇒14+34\Rightarrow \dfrac{1}{4} + \dfrac{3}{4}
Adding to get the final answer,
⇒44=1\Rightarrow \dfrac{4}{4} = 1

∴\therefore The value of sin⁡30∘.cos⁡60∘+cos⁡30∘.sin⁡60∘\sin 30^\circ .\cos 60^\circ + \cos 30^\circ .\sin 60^\circ =1 = 1.

Note: If you are good at remembering the formulas, then you must be able to identify that the given equation is just an expansion of one of the trigonometric formulas.
You must be aware of the formula sin⁡(a+b)=sin⁡a.cos⁡b+cos⁡a.sin⁡b\sin \left( {a + b} \right) = \sin a.\cos b + \cos a.\sin b. If you look closely on the given equation, you will observe that it is forming the RHS of the formula.
Let a=30∘a = 30^\circ and b=60∘b = 60^\circ . Putting this in the LHS of the formula,
⇒sin⁡(30∘+60∘)=sin⁡90∘\Rightarrow \sin \left( {30^\circ + 60^\circ } \right) = \sin 90^\circ
We can see in the table above what is the value of the trigonometric ratio. Therefore, sin⁡90∘=1\sin 90^\circ = 1.
Hence, the answer from both the methods is the same.