Question

Solve: $\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ }$

Answer 1

Hint: We will be using the addition formula of trigonometry to solve this question. The addition formula is $\sin A\cos B + \sin B\cos A = \sin (A + B)$ .

Complete step-by-step answer:
Method $1$:
We have to solve $\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ }$
Using the formula $\sin A\cos B + \sin B\cos A = \sin (A + B)$ , we get $A = {60^ \circ }$ and $B = {30^ \circ }$ .
Substituting these values,
$\sin A\cos B + \sin B\cos A = \sin (A + B)$
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \sin ({60^ \circ } + {30^ \circ })$
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \sin ({90^ \circ })$
We know, $\sin {90^ \circ } = 1$,
$\therefore \sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = 1$

Method $2$:
Standard trigonometric values:

	${0^ \circ }$	${30^ \circ }$	${45^ \circ }$	${60^ \circ }$	${90^ \circ }$
sin$\theta $	$0$	$\dfrac{1}{2}$	$\dfrac{1}{{\sqrt 2 }}$	$\dfrac{{\sqrt 3 }}{2}$	$1$
cos$\theta $	$1$	$\dfrac{{\sqrt 3 }}{2}$	$\dfrac{1}{{\sqrt 2 }}$	$\dfrac{1}{2}$	$0$
tan$\theta $	$0$	$\dfrac{1}{{\sqrt 3 }}$	$1$	$\sqrt 3 $	$\infty $

The value of $\sin {60^ \circ } = \dfrac{{\sqrt 3 }}{2}$ , $\cos {60^ \circ } = \dfrac{1}{2}$, $\sin {30^ \circ } = \dfrac{1}{2}$, $\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$
Substituting all these values in $\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ }$,
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \dfrac{{\sqrt 3 }}{2} \times \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2} \times \dfrac{1}{2}$
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \dfrac{3}{4} + \dfrac{1}{4}$
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \dfrac{{3 + 1}}{4}$
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \dfrac{4}{4}$
$\sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = \dfrac{4}{4}$
$\therefore \sin {60^ \circ }\cos {30^ \circ } + \sin {30^ \circ }\cos {60^ \circ } = 1$

Note: Angle sum identities and angle difference identities can be used to find the function values of any angles however, the most practical use is to find exact values of an angle that can be written as a sum or difference using the familiar values for the sine, cosine, and tangent of the ${0^ \circ }$, ${30^ \circ }$, ${45^ \circ }$, ${60^ \circ }$and ${90^ \circ }$ angles and their multiples.