Math Doubts

$\sin{15^\circ}$ value

Formula

$\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}-1}{2\sqrt{2}}$

The value of sine in a fifteen degrees right triangle is called the sine of angle fifteen degrees.

Introduction

sin 15 degrees

The sine of angle fifteen degrees is a value that represents the ratio of length of opposite side to the length of the hypotenuse when the angle of a right-angled triangle is fifteen degrees. According to the Sexagesimal angle measuring system, the sine of angle fifteen degrees is written as $\sin{(15^\circ)}$ in mathematical form.

Fractional value

$\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}-1}{2\sqrt{2}}$

The exact value of sine of angle fifteen degrees in fraction form is square root of three minus one divided by two times square root of two. The fractional value for sine of angle fifteen degrees is also written as follows.

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $1 \times \dfrac{\sqrt{3}-1}{2\sqrt{2}}$

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2}}{\sqrt{2}} \times \dfrac{\sqrt{3}-1}{2\sqrt{2}}$

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2} \times (\sqrt{3}-1)}{\sqrt{2} \times 2\sqrt{2}}$

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2} \times \sqrt{3}-\sqrt{2} \times 1}{\sqrt{2} \times 2 \times \sqrt{2}}$

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2 \times 3}-\sqrt{2}}{2 \times \sqrt{2} \times \sqrt{2}}$

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{6}-\sqrt{2}}{2 \times (\sqrt{2})^2}$

$\implies$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{6}-\sqrt{2}}{2 \times 2}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{6}-\sqrt{2}}{4}$ $\,=\,$ $\dfrac{\sqrt{3}-1}{2\sqrt{2}}$

Therefore, the value of sine of fifteen degrees in fraction is exactly equal to the square root of six minus square of two divided by four.

Approximate value

$\sin{(15^\circ)}$ $\,=\,$ $0.2588190451\cdots$

The sine of angle fifteen degrees is an irrational number. Its value in decimal form is given here and its approximate value is $0.2588$.

$\implies$ $\sin{(15^\circ)}$ $\,\approx\,$ $0.2588$

Other forms

In trigonometry, the sine of angle $15$ degrees is written alternatively in two popular forms.

Circular system

$\sin{\Big(\dfrac{\pi}{12}\Big)}$ $\,=\,$ $\dfrac{\sqrt{3}-1}{2\sqrt{2}}$

The sine of $15^\circ$ is written as the sine of angle pi divided by twelve radians in circular system and it is expressed as $\sin{\Big(\dfrac{\pi}{12}\Big)}$ in mathematical form.

Centesimal system

$\sin{\Big(16\small\dfrac{2}{3}^{\normalsize g\,}\normalsize\Big)}$ $\,=\,$ $\dfrac{\sqrt{3}-1}{2\sqrt{2}}$

The sine of angle fifteen degrees is also written as sine of angle sixteen-two divided by three gradians and it is written as $\sin{\Big(16\small\dfrac{2}{3}^{\normalsize g\,}\normalsize\Big)}$ in mathematical form as per the centesimal system.

Proofs

Learn how to prove the exact value of sine of angle fifteen degrees in both fraction and approximate forms as per the trigonometric and geometric methods.

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