Determine the value of sin 39°

Finding the value of \sin 39° 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 39° is an irrational number.

Approximate value of sin 39°

\sin 39°\approx 0.629320391

Once the value of \cos 75° is available, it's easy to obtain the value of \cos 51 ° using the trigonometric co-function identity.

\cos 51 °=\cos(90° - 39°)=\sin 39°

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

\sin \dfrac{13\pi}{60} = \sin 39°

A question may arise, how does the approximate value come out? Is it possible to derive the exact value of \sin 39° ?

Well, since 39\degree is a multiple of 3 , \sin 39\degree could be derived by using trigonometric identities and values of special angles.

39\degree = 75\degree -36\degree

Normally, there are more than one way to derive the trigonometric value of a special angle. The exact value of \sin 39\degree could be determined by the trigonometric values of 75\degree and 36\degree using difference identity for sin function. Since 39\degree is half of 78\degree, \sin 39\degree could also be derived by using half angle identity and trigonometric values of 78\degree.

Using difference identity for sine function

The difference identity for sine function could transform a trigonometric function of an angle to the difference of products of sine and cosine functions of two angles.

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

(1)

The exact value of \sin 39\degree could be determined by the trigonometric values of 75\degree and 36\degree using difference identity for sine function.

\sin(76°-36° ) = \sin 75°\cos 36° -\cos 75°\sin 36°

(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 \sin 75° &\dfrac{\sqrt{6}+\sqrt{2} }{4} \\ \hline \cos 75° &\dfrac{\sqrt{6}-\sqrt{2} }{4} \\ \hline \sin 36° &\dfrac{1}{4}\cdotp \sqrt{10-2\sqrt{5} } \\ \hline \cos 36° &=\dfrac{1+\sqrt{5} }{4} \\ \hline \end{array}

The exact values for \sin 75° and ° could be derived by trigonometric values of 30° and 45°. The exact values of \sin 36° and \cos 36° could be derived geometrically with the help of golden triangle.

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(75°-36° ) = \sin 75°\cos 36° -\cos 75°\sin 36°

=\dfrac{\sqrt{6}+\sqrt{2} }{4} \cdot \dfrac{1+\sqrt{5} }{4} - \dfrac{\sqrt{6}-\sqrt{2} }{4} \cdot \dfrac{1}{4}\cdotp \sqrt{10-2\sqrt{5} }

Therefore, rearrange the result and we get the exact value of \sin 39° is as follow

\dfrac{1}{16}[(\sqrt{6}+\sqrt{2} )(1+\sqrt{5} ) - (\sqrt{6}-\sqrt{2} ) \sqrt{10-2\sqrt{5} }]

Using the Microsoft Excel formula

=SIN(PI()*75/180)*(1+SQRT(5))/4-(SQRT(6)-SQRT(2))/4SQRT(10-2SQRT(5))/4

to verify it shows the result match the approximate value of 39°, which is 0.629320391

Collected in the board: Trigonometric Values of Special Angles

Steven Zheng posted 5 months ago