Determine the exact value of sin 57°

Finding the value of 57° 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 57° is an irrational number.

\sin 57°\approx 0.838670568

Once the value of \sin 57° is available, the trigonometric values of relevant angles could be obtained, for example, the value of \cos 36 ° using the trigonometric co-function identity.

\cos 33 °=\cos(90° - 57°)=\sin 57°

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

\sin \dfrac{19\pi}{60} = \sin 57°

A question arises, how does the approximate value come out? Is it possible to derive the exact value of \sin 57° ?

Since 57\degree is a multiple of 3 \degree , \sin57\degree could be derived by using trigonometric identities and values of special angles.

57\degree = 45\degree +12\degree

The exact value of \sin 57\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 12° & \dfrac{1}{8}(\sqrt{10+2\sqrt{5}} +\sqrt{3} -\sqrt{15} ) \\ \hline \cos12° & \dfrac{1}{8}(\sqrt{3} \sqrt{10+2\sqrt{5}} -1 +\sqrt{5} ) \\ \hline \end{array}

Trigonometric values of 45° angle can be obtained with the help of an isosceles right triangle. Trigonometric values of 12° could be derived with the help of trig values of special angles of 30\degree and 18\degree. The following two posts give details on the steps of the derivation of trig values of 12\degree.

Determine the value of sin12°

Determine the value of cos 12°

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° +12° ) = \sin 45°\cos 12°+\cos 45° \sin 12°

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

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

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

= \dfrac{\sqrt{2} }{16}(\sqrt{3} \sqrt{10+2\sqrt{5}} -1 +\sqrt{5} +\sqrt{10+2\sqrt{5}} +\sqrt{3} -\sqrt{15} )

Using the Microsoft Excel formula

=SQRT(2)/16*(SQRT(3)*SQRT(10+2SQRT(5))-1+SQRT(5)+SQRT(10+2SQRT(5))+SQRT(3)-SQRT(15))

to verify it shows the result match the approximate value of \sin 57°, which is 0.838670568

Collected in the board: Trigonometric Values of Special Angles

Steven Zheng posted 1 year ago