Question

Find the value of this and ....
${ \sin(45) }^{2} \times { \cos(30) }^{2} + \frac{1}{2} { \cos(90) }^{2}$
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Answer 1

Trigonometry

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:

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

Solution:

We are asked to find the exact value of the given equation:

$\longrightarrow {\sin^2(45)} \times {\cos^2(30)} + \dfrac{1}{2} {\cos^2(90)}$

By substituting the known values of trigonometric angles in the equation, we get the following results:

$\implies \bigg(\dfrac{1}{\sqrt{2}}\bigg)^2 \times \bigg(\dfrac{\sqrt{3}}{2}\bigg)^2 + \dfrac{1}{2} \times (0)^2$

$\implies \dfrac{(1)^2}{(\sqrt{2})^2} \times \dfrac{(\sqrt{3})^2}{(2)^2} + \dfrac{1}{2} \times 0$

$\implies \dfrac{1}{2} \times \dfrac{3}{4} + \dfrac{1}{2} \times 0$

$\implies \dfrac{1 \times 3}{2 \times 4} + \dfrac{1}{2} \times 0$

$\implies \dfrac{3}{8} + \dfrac{1}{2} \times 0$

$\implies \dfrac{3}{8} + 0$

$\implies \dfrac{3}{8}$

Therefore the required answer is:

$\boxed{{\sin^2(45)} \times {\cos^2(30)} + \dfrac{1}{2} {\cos^2(90)} = \dfrac{3}{8}}$

To read and understand some similar type of questions from this chapter, refer the below link:

brainly.in/question/48030979