Math Doubts

Proof of $\cos{(18^°)}$ value

The exact value of cos of 18 degrees can be derived mathematically in trigonometry. It can be calculated in mathematics theoretically and also geometrically. So, let us learn both methods to derive the $\cos{(18^°)}$ mathematically.

Trigonometric approach

Theoretically, it is not possible to find the exact value of cosine of $18$ degrees in fraction from directly but it can be evaluated from the value of the sine of angle $18$ degrees.

Expression of Cosine in terms of sine function

According to the transformation identity of cosine in terms of sine function, the value of cosine can be evaluated from the value of sine function.

$\cos{\theta} \,=\, \sqrt{1-\sin^2{\theta}}$

Now, take the angle $\theta$ is $18$ degrees.

$\implies$ $\cos{(18^°)} = \sqrt{1-\sin^2{(18^°)}}$

Evaluate sine of angle 18 degrees

Now, we have to evaluate the sin of 18 degrees. So, let us derive the exact value of sin of 18 degrees mathematically.

$\sin{(18^°)} \,=\, \dfrac{\sqrt{5}-1}{4}$

Evaluate Cosine of angle 18 degrees

Now, substitute the value of sine of $\dfrac{\pi}{10}$ radians in the equation, which expresses the cosine in terms of sine function.

$\implies$ $\cos{(18^°)}$ $=$ $\sqrt{1-{\Bigg(\dfrac{\sqrt{5}-1}{4}\Bigg)}^2}$

Let’s try to simplify the value of cosine of $20$ gradians.

$=\,\,\,$ $\sqrt{1-\dfrac{{(\sqrt{5}-1)}^2}{4^2}}$

$=\,\,\,$ $\sqrt{1-\dfrac{{(\sqrt{5})}^2+{(1)}^2-2 \times \sqrt{5} \times 1}{16}}$

$=\,\,\,$ $\sqrt{1-\dfrac{5+1-2\sqrt{5}}{16}}$

$=\,\,\,$ $\sqrt{1-\dfrac{6-2\sqrt{5}}{16}}$

$=\,\,\,$ $\sqrt{\dfrac{16-(6-2\sqrt{5})}{16}}$

$=\,\,\,$ $\sqrt{\dfrac{16-6+2\sqrt{5}}{16}}$

$=\,\,\,$ $\sqrt{\dfrac{10+2\sqrt{5}}{16}}$

$=\,\,\,$ $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\cos{(18^°)}$ $\,=\,$ $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$

Therefore, we have derived the exact value of cosine of angle $18$ degrees in fraction and its value can be calculated mathematically but it is an irrational number.

$\implies$ $\cos{(18^°)}$ $\,=\,$ $0.9510565162\ldots$

$\implies$ $\cos{(18^°)}$ $\,\approx\,$ $0.9511\ldots$

Geometrical approach

The cosine of angle $18$ degrees can also derived geometrically by constructing a right triangle with an angle $18^°$.

construction of right angled triangle with 18 degrees angle
  1. Use a ruler and draw a line of any length horizontally. For example, $10 \, cm$ line is drawn and it is called the line $\overline{DE}$.
  2. Use a protractor and draw a perpendicular line to the line segment $\overline{DE}$ at point $E$.
  3. Now, coincide the middle point of the protractor with the point $D$, then mark on plane at $18$ degrees indication line of protractor in anticlockwise direction. Finally, draw a line from point $D$ through $18$ degrees mark and it intersects the perpendicular line at point $F$.

The three steps helped us in constructing a right triangle, known as $\Delta FDE$. In this case, the angle of the right angled triangle is $18$ degrees. So, let us evaluate the cosine of angle $\dfrac{\pi}{10}$ radian.

$\cos{(18^°)} \,=\, \dfrac{DE}{DF}$

The length of the adjacent side ($\overline{DE}$) is $10 \, cm$ but the length of the hypotenuse ($\overline{DF}$) is unknown. However, it can be measured by using a ruler and it is measured that the length of the hypotenuse is $10.5 \, cm$.

$\implies$ $\cos{(18^°)} \,=\, \dfrac{10}{10.5}$

$\implies$ $\cos{(18^°)} \,=\, 0.9523809523\ldots$

Verdict

Let us discuss about the value of cosine of angle $18$, obtained from the theoretical and geometrical approaches.

  1. Theoretically, it is derived that the value of $\cos{(18^°)}$ is $0.9510565162\ldots$
  2. Geometrically, it is derived that the value of $\cos{(18^°)}$ is $0.9523809523\ldots$

It can be observed that the values of cosine of angle $20$ gradians, obtained from both theoretical and geometrical methods are different. The $\cos{(18^°)}$ value that obtained from theoretical method is accurate but the $\cos{(20^g)}$ value that obtained from geometrical method is approximate because the length of hypotenuse is measured approximately.

$( 0.9510565162\ldots )$ $\,-\,$ $( 0.9523809523\ldots )$ $\,=\,$ $-0.0013244360\ldots$

$\implies$ $( 0.9510565162\ldots )$ $\,-\,$ $( 0.9523809523\ldots )$ $\,\approx\,$ $0$

However, the difference between them is $-0.0013244360\ldots$, which is small, negligible and approximately zero.

Therefore, it is verified that the exact value of $\cos{(18^°)}$ is $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$ in fraction form and $0.9510565162\ldots$ in decimal form.

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