Proof for Sine of 75 degrees in Trigonometric method

The exact value of sine of angle seventy five degrees in fraction form is equal to the quotient of square root of six plus square root of two by four.

$\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{6}+\sqrt{2}}{4}$

The sine of angle five pi by twelve radian can be proved exactly in fraction form in trigonometric mathematics. So, let us learn how to derive the sine seventy five degrees in trigonometry.

Angle sum identity of sine function

The angle $75^\circ$ can be written as the sum of two known angles $30^\circ$ and $45^\circ$. Hence, the sine of angle $75$ degrees can be written as the sine of sum of them.

$\sin{(75^\circ)}$ $\,=\,$ $\sin{(45^\circ+30^\circ)}$

The sine of sum of the angles $45^\circ$ and $30^\circ$ can be expanded by the sine angle sum identity.

$\sin{(x+y)}$ $\,=\,$ $\sin{x}\cos{y}$ $+$ $\cos{x}\sin{y}$

Let $x \,=\, 45^\circ$ and $y \,=\, 30^\circ$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\sin{(45^\circ+30^\circ)}$ $\,=\,$ $\sin{(45^\circ)}\cos{(30^\circ)}$ $+$ $\cos{(45^\circ)}\sin{(30^\circ)}$

Substitute values of sine and cosine functions

According to the trigonometry, the exact values of sin 30 degrees, sin 45 degrees, cos 30 degrees and cos 45 degrees are already known to us.

$(1).\,\,\,$ $\sin{(45^\circ)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$

$(2).\,\,\,$ $\cos{(30^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}}{2}$

$(3).\,\,\,$ $\cos{(45^\circ)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$

$(4).\,\,\,$ $\sin{(30^\circ)}$ $\,=\,$ $\dfrac{1}{2}$

Now, we substitute them in the expansion of the sine of sum of angles $30$ and $45$ degrees expression.

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\sin{(45^\circ+30^\circ)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2}$ $+$ $\dfrac{1}{\sqrt{2}} \times \dfrac{1}{2}$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2}$ $+$ $\dfrac{1}{\sqrt{2}} \times \dfrac{1}{2}$

Evaluate the sine 75 degrees by simplification

The sine of angle $75$ degrees is expressed as a mathematical expression in numerical form. In order to calculate it, we have to simplify the right hand expression of the equation.

$\sin{(75^\circ)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{3}}{2}$ $+$ $\dfrac{1}{\sqrt{2}} \times \dfrac{1}{2}$

Each term in the right hand side expression of the equation represents the multiplication of two fractions. So, let’s evaluate the product of the fractions for each term by the multiplication of the fractions.

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{1 \times \sqrt{3}}{\sqrt{2} \times 2}$ $+$ $\dfrac{1 \times 1}{\sqrt{2} \times 2}$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{1 \times \sqrt{3}}{2 \times \sqrt{2}}$ $+$ $\dfrac{1 \times 1}{2 \times \sqrt{2}}$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}}{2\sqrt{2}}$ $+$ $\dfrac{1}{2\sqrt{2}}$

The right hand side expression expresses an addition of the two like fractions. So, we can find their sum by the addition of the like fractions.

$\,\,\,\therefore\,\,\,\,\,\,$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}+1}{2\sqrt{2}}$

It can be further simplified by eliminating the square root of $2$ from the denominator of the right hand side expression of the equation.

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}+1}{2\sqrt{2}}$ $\times$ $1$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}+1}{2\sqrt{2}}$ $\times$ $\dfrac{\sqrt{2}}{\sqrt{2}}$

Now, multiply the fractions for simplifying the mathematical expression of sine of $5\pi$ by $12$.

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{(\sqrt{3}+1) \times \sqrt{2}}{2\sqrt{2} \times \sqrt{2}}$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2} \times (\sqrt{3}+1)}{2 \times \sqrt{2} \times \sqrt{2}}$

In the denominator, the factors can be expressed in exponential notation.

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2} \times (\sqrt{3}+1)}{2 \times (\sqrt{2})^2}$

In the numerator, the multiplication of square root of $2$ can be distributed to the sum of the square root of $3$ and one by the distributive property of multiplication over addition.

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2} \times \sqrt{3}+ \sqrt{2} \times 1}{2 \times 2}$

$\implies$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{2 \times 3}+\sqrt{2}}{4}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\sin{(75^\circ)}$ $\,=\,$ $\dfrac{\sqrt{6}+\sqrt{2}}{4}$

Therefore, the exact value for the sine of angle seventy five degrees in fraction form is equal to the quotient of square root of six plus square root of two by four.