Chapter 6, Problem 12CT

To determine

The exact value of the expression, $2{\sin }^2{60}^0-3\cos {45}^0$.

Answer to Problem 12CT

Solution:

The exact value of the expression $2{\sin }^2{60}^0-3\cos {45}^0$ is $\frac{3\left(1-\sqrt{2}\right)}{2}$.

Explanation of Solution

Given information:

The expression, $2{\sin }^2{60}^0-3\cos {45}^0$

Explanation:

Let, the expression be $2{\sin }^2{60}^0-3\cos {45}^0$.

To determine exact value of the expression, $2{\sin }^2{60}^0-3\cos {45}^0$, first find out the exact value of $\sin \left({60}^0\right)$ and $\cos \left({45}^0\right)$ using unit circle.

The point $P$ on the unit circle that corresponds to $\theta ={60}^0$ is $P=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$,

So, $\sin {60}^0=y=\frac{\sqrt{3}}{2}$.

Then,

$2{\sin }^2{60}^0=2{\left(\frac{\sqrt{3}}{2}\right)}^2=2\left(\frac{3}{4}\right)=\frac{3}{2}$

The point $P$ on the unit circle that corresponds to $\theta ={45}^0$ is $P=\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$, so $\cos {45}^0=x=\frac{\sqrt{2}}{2}$.

Thus, $2{\sin }^2{60}^0-3\cos {45}^0=\frac{3}{2}-3\left(\frac{\sqrt{2}}{2}\right)=\frac{3\left(1-\sqrt{2}\right)}{2}$

Thus, the exact value of the expression $2{\sin }^2{60}^0-3\cos {45}^0$ is $\frac{3\left(1-\sqrt{2}\right)}{2}$.