Find the exact value of sin 42°

Finding the value of 42° 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 42° is an irrational number.

\sin 42°\approx 0.669130606

Once the value of \sin 42° is available, it's easy to obtain the value of \cos 48 ° using the trigonometric co-function identity.

\cos 48 °=\cos(90° - 42°)=\sin 42°

Since there are two units used for angle measurement, degree and radian, \sin 42° could also be expressed in radians angle.

\sin \dfrac{7\pi}{30} = \sin 42°

A question may arise, how does the approximate value come out? Is it possible to derive the exact value of \sin 42° ?

Well, since 42\degree is a multiple of 3 \degree , \sin42\degree could be derived by using trigonometric identities and values of special angles.

42\degree = 60\degree -18\degree

The exact value of \sin 42\degree could be determined by the trigonometric values of 60\degree and 18\degree using difference 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 60° & \dfrac{\sqrt{3} }{2} \\ \hline \cos 60° & \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 60° and \cos 60° 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(60° -18° ) = \sin 60°\cos 18° -\cos 60° \sin 18°

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

Therefore, rearrange the result and we get the exact value of \sin 42° is as follow

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

Using the Microsoft Excel formula

=(SQRT(3)*SQRT(10+2*SQRT(5))+1-SQRT(5))/8

to verify it shows the result match the approximate value of \sin 42°, which is 0.669130606.

Collected in the board: Trigonometry

Steven Zheng posted 1 year ago