Question

Simplify sin30° cos45° + cos30° sin45°

Answer 1

Trigonometry

We have been asked to find the exact value of the following trigonometric equation:

$\longrightarrow \sin(30^{\circ}) \cos(45^{\circ}) + \cos(30^{\circ}) \sin(30^{\circ})$

Pre-requisite knowledge:

The following are the tips and concept that can be use to find the solution:

• Having a basic knowledge of Trigonometric Angles.

• Trigonometric ratios are sine, cosine, tangent, cosecant, secant, cotangent.

• The standard angles of these trigonometric ratios are 0°, 30°, 45°, 60° and 90°.

Analyse the values of important angles for all the six trigonometric ratios shown in the table given below:

$\boxed{\begin{array}{c|c|c|c|c|c}\bf x & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \sin(x) & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \cos(x) & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \tan(x) & 0 & \dfrac{1}{ \sqrt{3}} &1 & \sqrt{3} & \rm \infty \\ \\ \csc(x) & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \sec(x) & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \cot(x) & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{\sqrt{3}} & 0\end{array}}$

Step-by-step solution:

By substituting the known values of trigonometric angles in the equation, the following results are obtained:

$\implies \sin(30^{\circ}) \cos(45^{\circ}) + \cos(30^{\circ}) \sin(30^{\circ})$

$\implies \dfrac{1}{2} \times \dfrac{1}{\sqrt{2}} + \dfrac{\sqrt{3}}{2} \times \dfrac{1}{\sqrt{2}}$

$\implies \dfrac{1 \times 1}{2 \times \sqrt{2}} + \dfrac{\sqrt{3} \times 1}{2 \times \sqrt{2}}$

$\implies \dfrac{1}{2\sqrt{2}} + \dfrac{\sqrt{3}}{2\sqrt{2}}$

$\implies \dfrac{\sqrt{2}}{4} + \dfrac{\sqrt{6}}{4}$

$\implies \dfrac{\sqrt{2} + \sqrt{6}}{4}$

Therefore, the required answer is:

$\boxed{\sin(30^{\circ}) \cos(45^{\circ}) + \cos(30^{\circ}) \sin(30^{\circ}) = \dfrac{\sqrt{2} + \sqrt{6}}{4}}$

Answer 2

TO SOLVE :

$\rm⇢ \: \: \sin(30 \degree) . \cos(45 \degree) + \cos(30 \degree). \sin(30 \degree)$

SOLUTION :-

Let's solve this by using the Trigonometric Values of Trigonometric Ratios .

The Values of Important Angles for all six Trigonometric Ratios listed in the Table Given Below :-

$\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} }, & 0 \end{array}}}$

then, Substituting the values, we get

$\rm⇢ \: \: (\frac{1}{2} \times \frac{1}{ \sqrt{2} } ) + ( \frac{ \sqrt{3} }{2} \times \frac{1}{ \sqrt{2} } )$

$\rm⇢ \: \: ( \frac{1}{2 \sqrt{2} } ) + ( \frac{ \sqrt{3} }{2 \sqrt{2} } )$

$\rm⇢ \: \: (\frac{ \sqrt{2} }{4}) + ( \frac{ \sqrt{6} }{4} )$

$\rm⇢ \: \: \frac{ \sqrt{2} + \sqrt{6} }{4}$