Show $\sin(\frac{\pi}{3})=\frac{1}{2}\sqrt{3}$

Question

I have to show that

$$\sin\left(\frac{\pi}{3}\right)=\frac{1}{2}\sqrt{3}$$

and

$$\cos\left(\frac{\pi}{3}\right)=\frac{1}{2}$$

Should I use the exponential function?

Answer 1

Hint. From the picture below and Pythagoras we get

$$h=\frac{\sqrt 3}{2}a\rightarrow \sin\frac\pi 3=\frac ha=\frac{\frac{\sqrt 3}{2}a}{a}=\frac{\sqrt 3}{2}.$$

Answer 2

consider an equilateral triangle and and construct one hight of this triangle then we have a right triangle and we get $\sin(\pi/3)=\frac{h}{a}$ with $h=\frac{\sqrt{3}}{2}a$ we get the searched term, where a is the side length of the triangle

Answer 3

Hint:

It's a simple geometric result. See the figure and note that the triangle $OPM$ is equilateral.

If $OP =1$ than $\sin (\pi/3)=PH$ and $\cos( \pi/3)=OH$.

Answer 4

Consider an equilateral triangle of side length $a$. We will compute the area by two different ways to compute the value of $\sin(\pi/3)$.

• Method $1$: Since the semi-perimeter is $s = \dfrac{3a}2$, using Heron's formula we have the area of the triangle to be $\dfrac{\sqrt3}4a^2$
• Method $2$: The height of the altitude is $a\sin(\pi/3)$ and hence the area is $\dfrac12 \cdot a \cdot a\sin(\pi/3)$.

Comparing the two, we obtain that $\sin(\pi/3) = \sqrt{3}/2$

Answer 5

Note that:

$sin(3\theta)=3sin(\theta)-4sin^{3}(\theta)$

Let $\theta =\frac{\pi}{3}$ and let $sin(\frac{\pi}{3})=x$

Then:

$0=x(3-4x^{2})$.

Hence either $x=0$ or $x^{2}=\frac{3}{4}$

But since $sin(0)=0$ and the $sine$ function has a period of $2\pi$ then we must conclude that $sin(\frac{\pi}{3})=\frac{\sqrt3}{2}$

Answer 6

Not to confuse you. But.. $$x=\dfrac{\pi}{3}$$ $$3x=\pi$$ $$\sin 3x=\sin \pi$$ $$3\sin x-4\sin^3x=0$$ $$\sin x=0 \text{ or } \sin x = \dfrac{\sqrt{3}}{2} \text{ or } \sin x = \dfrac{-\sqrt{3}}{2}$$

$$\text{as } 0<\dfrac{\pi}{3}<\dfrac{\pi}{2} \text{, } \sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$$

Answer 7

Hint: a the sum of the angles of a triangle is $180^\circ$ (or $\pi$ radians). If all of the angles were equal, what would the angle measure of each angle be? Can you find its altitude?