Determine the exact value of sin 63°

Finding the value of 63° 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 63° is an irrational number.

\sin 63°\approx 0.891006524

Once the value of \sin 63° is available, the trigonometric values of relevant angles could be obtained, for example, the value of \cos 27 ° using the trigonometric co-function identity.

\cos 27 °=\cos(90° - 63°)=\sin 63°

Since there are two units used for angle measurement, degree and radian, \sin 57° could also be expressed in radians angle.

\sin \dfrac{7\pi}{20} = \sin 63°

A question arises, how does the approximate value come out? Is it possible to derive the exact value of \sin 63° ?

Since 63\degree is a multiple of 3 \degree , \sin63\degree could be derived by using trigonometric identities and values of special angles.

63\degree = 45\degree +18\degree

The exact value of \sin 63\degree could be determined by the trigonometric values of 45\degree and 12\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 45° & \dfrac{ \sqrt{2} }{2} \\ \hline \cos 45° & \dfrac{ \sqrt{2} }{2} \\ \hline \sin 18° & \dfrac{-1+\sqrt{5} }{4} \\ \hline \cos18° &\dfrac{\sqrt{10+2\sqrt{5} } }{4} \\ \hline \end{array}

Trigonometric values of 45° angle can be obtained with the help of an isosceles right triangle. Trigonometric values of 18° could be derived with the help of golden triangle. The following two posts give details on the steps of the derivation of trig values of 18\degree.

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(45° +18° ) = \sin 45°\cos 18°+\cos 45° \sin 18°

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

Therefore, rearrange the result and we get the exact value of \sin 63° as follow

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

Using the Microsoft Excel formula

=SQRT(2)/8*(SQRT(10+2*SQRT(5))-1+SQRT(5))

to verify it shows the result match the approximate value of \sin 63°, which is 0.891006524

Collected in the board: Trigonometric Values of Special Angles

Steven Zheng posted 1 year ago