Find the exact value of sin 12°

Finding the value of 12° 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 12° is an irrational number.

Approximate value of sin 12°

\sin 12°\approx 0.207911691

Once the value of \sin 12° is available, it's easy to obtain the value of \cos 78 ° using the trigonometric co-function identity.

\cos 78 °=\cos(90° - 12°)=\sin 12°

Since there are two units used for angle measurement, degree and radian, \sin 12° could also be expressed in radians angle.

\sin \dfrac{\pi}{15} = \sin 12°

A question may arise, how does the approximate value come out? Is it possible to derive the exact value of \sin 12° ?

Well, since 12\degree is a multiple of 3 , \sin12\degree could be derived by using trigonometric identities and values of special angles.

12\degree = 30\degree -18\degree

Normally there are more than one way to derive the trigonometric value of a special angle. The exact value of \sin 12\degree could be determined by the trigonometric values of 30\degree and 18\degree using difference identity for sines function.

Using difference identity for sines function

The difference identity could transform a trigonometric function of an angle to the difference of products of trigonometric functions of two angles that are the difference of the original angle.

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

(1)

The exact value of \sin 12\degree could be determined by the trigonometric values of 30\degree and 18\degree using difference identity for sines function.

\sin(30°-12° ) = \sin 30°\cos 18°-\cos 30 \sin 18°

(2)

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 \cos 30° & \dfrac{\sqrt{3} }{2} \\ \hline \sin 30° & \dfrac{1 }{2} \\ \hline \sin 18° & \dfrac{-1+\sqrt{5} }{4} \\ \hline \cos18° &\dfrac{\sqrt{10+2\sqrt{5} } }{4} \\ \hline \end{array}

The exact values for \sin 30° and \cos 30° could be determined with the help of 30-60-90 right angle triangle. The exact values of \sin 18° and \cos 18° 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°

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(30° -18° ) = \sin 30°\cos 18°-\cos 30° \sin 18°

= \dfrac{1 }{2}\cdot \dfrac{\sqrt{10+2\sqrt{5} } }{4} - \dfrac{\sqrt{3} }{2} \cdot \dfrac{-1+\sqrt{5} }{4}

Therefore, rearrange the result and we get the exact value of \sin 12° is as follow

\dfrac{1}{8}(\sqrt{10+2\sqrt{5}} +\sqrt{3} -\sqrt{15} )

Using the Microsoft Excel formula

=(SQRT(10+2*SQRT(5))+SQRT(3)-SQRT(15))/8

to verify it shows the result match the approximate value of \sin 12°, which is 0.207911691.

Collected in the board: Trigonometric Values of Special Angles

Steven Zheng posted 5 months ago