Determine the exact value of sin 54°

Finding the value of 54° could be achieved by looking up trigonometric table or simply calculating using various calculator. An approximate decimal value is given since the exact value of \sin 54° is an irrational number.

\sin 54°\approx 0.809016994

Once the value of \sin 54° is available, the trigonometric values of relevant angles could be obtained, for example, the value of \sin 36 ° using the trigonometric co-function identity.

\cos 36 °=\cos(90° - 54°)=\sin 54°

Since there are two units used for angle measurement, degree and radian, \sin 54° could also be expressed in radians angle.

\sin \dfrac{3\pi}{10} = \sin 54°

A question may arise, how does the approximate value come out? Is it possible to derive the exact value of \sin 54° ?

Well, since 54\degree is a multiple of 3 \degree , \sin54\degree could be derived by using trigonometric identities and values of special angles.

54\degree = 36\degree +18\degree

The exact value of \sin 54\degree could be determined by the trigonometric values of 36\degree and 18\degree using sum identity for sines function.

\sin(\alpha +\beta ) = \sin \alpha\cos \beta +\cos \alpha \sin \beta

(1)

The trigonometric values of special angles we are going to use are listed in the table below.

\def\arraystretch{2.4} \begin{array}{|c|c|} \hline \sin 36° & \dfrac{ \sqrt{10-2\sqrt{5} } }{4} \\ \hline \cos 36° & \dfrac{1+\sqrt{5} }{4} \\ \hline \sin 18° & \dfrac{-1+\sqrt{5} }{4} \\ \hline \cos18° &\dfrac{\sqrt{10+2\sqrt{5} } }{4} \\ \hline \end{array}

Both 18° and 36° are special angers that are related to golden triangle. An angle of 36° could be th vertex angle of the golden isosceles triangle while an angle of 18° is the half angle of the vertex angle. Therefore, the exact trigonometric values of 18° and 36° could be derived geometrically with the help of golden triangle or by using triple angle identities. The following two posts give details on the steps of their derivation.

Determine the value of cos⁡18°

Determine the value of sin⁡18°

Determine the value of sin⁡36°

Determine the value of cos 36°

It should be noted that most of the trigonometric values of these special angles are irrational numbers. So it is expected the same for the values that are resulted in by operation of these numbers.

Now let's substitute special angles and their trigonometric values to the identity (1)

\sin(36° +18° ) = \sin 36°\cos 18°+\cos 36° \sin 18°

=\dfrac{ \sqrt{10-2\sqrt{5} } }{4} \cdot \dfrac{\sqrt{10+2\sqrt{5} } }{4} + \dfrac{1+\sqrt{5} }{4} \cdot \dfrac{-1+\sqrt{5} }{4}

Therefore, rearrange the result and we get the exact value of \sin 54° as follow

\dfrac{1+\sqrt{5} }{4}

Using the Microsoft Excel formula

=(1+SQRT(5))/4

to verify it shows the result match the approximate value of \sin 54°, which is 0.809016994.

Collected in the board: Trigonometric Values of Special Angles

Steven Zheng posted 1 year ago