Question

Solve the following equation using trigonometric identities:
$\sin 60{}^\circ \cos 30{}^\circ +\sin 30{}^\circ \cos 60{}^\circ $

Answer 1

Hint: When we see the terms of the equation are sin, cos, tan you must use the properties of trigonometry. If there are variables then we use formula. But, here we have the values in degrees. As you know the values of angle 0,30,60,90 generally, for all trigonometric terms like sin, cos, tan. You can just substitute these values and find the final result. By using general algebra for more clarity substitute the values one by one. After substituting all values you will be left with nothing but normal mathematical expression with operations of sum, difference, and product. So, you can solve it with basic calculations. The result you get after calculation will be our answer.

Complete step-by-step solution -
Trigonometric Expressions: If an expression is written in terms of sin, cos, cot, tan, etc, then the expression is called a trigonometric expression. To solve them you can use general trigonometric relations or you can also use the values of general angles you know before-hand.
By general basic knowledge of trigonometry, we say the values:
$\sin 30{}^\circ =\dfrac{1}{2}$ ………………………….(1)
$\cos 30{}^\circ =\dfrac{\sqrt{3}}{2}$ ……………………(2)
$\sin 60{}^\circ =\dfrac{\sqrt{3}}{2}$ ……………………….(3)
$\cos 60{}^\circ =\dfrac{1}{2}$ ……………………………..(4)
Given expression in the question in terms of $\sin ,\cos $ is written as:
$\sin 60{}^\circ \cos 30{}^\circ +\sin 30{}^\circ \cos 60{}^\circ $ …………………………………(5)
By substituting equation (1) in equation (5), we get the equation as:
$\Rightarrow \sin 60\cos 30+\left( \dfrac{1}{2} \right)\cos 60$
By substituting equation (2) in above equation, we get it as:
$\Rightarrow \sin 60\left( \dfrac{\sqrt{3}}{2} \right)+\left( \dfrac{1}{2} \right)\cos 60$
By substituting equation (3) in above equation, we get it as:
$\Rightarrow \left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{\sqrt{3}}{2} \right)+\left( \dfrac{1}{2} \right)\cos 60$
By substituting equation (4) in above equation, we get it as:
$\Rightarrow \left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{\sqrt{3}}{2} \right)+\left( \dfrac{1}{2} \right)\left( \dfrac{1}{2} \right)$
By simplifying both the products, we can write it as:
$\Rightarrow \dfrac{3}{4}+\dfrac{1}{4}$
By taking least common multiple, we can write it as: $\dfrac{4}{4}$
By cancelling common terms in numerator, denominator we get it as: 1 .
Therefore, the value of $\sin 60{}^\circ \cos 30{}^\circ +\sin 30{}^\circ \cos 60{}^\circ $ is 1.

Note: Be careful while substituting the values as if you substitute in the wrong place you might get the wrong answer. Alternate method is to use a trigonometric formula given by $\sin A\cos B+\cos A\sin B=\sin \left( A+B \right)$ . Here, $A=60,B=30$ so, we get the result as $\sin \left( 60+30 \right)=\sin 90$ we know its value is 1. So, we got the same result in this method.