Find the exact value of cos 75°

Finding the value of \cos 75° 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 \cos 75° is an irrational number.

Approximate value of cos 75°

\cos 75°\approx 0.258819045

Once the value of \cos 75° is available, it's easy to obtain the value of \sin 15 ° using the trigonometric co-function identity.

\sin 15 °=\sin(90° - 75°)=\cos 75°

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

\cos \dfrac{5\pi}{12} = \cos 75°

A question may arise, how does the approximate value come out? Is it possible to derive the exact value of \cos 75° ?

Well, since 75\degree is a multiple of 3 , \cos 75\degree could be derived by using trigonometric identities and values of special angles.

75\degree = 45\degree +30\degree

Normally there are more than one way to derive the trigonometric value of a special angle. The exact value of \cos 75\degree could be determined by the trigonometric values of 30\degree and 45\degree using sum identity for cosine function. Since 75\degree is half of 150\degree, \cos 75\degree could also be derived by using half angle identity.

Using sum identity for cosine function

The sum identity for cosine function could transform a trigonometric function of an angle to the difference of products of cosine functions of two angles and products of sines functions of two angles that are added up to the original angle.

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

(1)

The exact value of \cos 75\degree could be determined by the trigonometric values of 30\degree and 45\degree using sum identity for cosine function.

\cos(30°+45° ) = \cos 30°\cos 45° -\sin 30°\sin 45°

(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 45° & \dfrac{\sqrt{2} }{2} \\ \hline \cos45° &\dfrac{\sqrt{2 } }{2} \\ \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 45° and \cos 45° could be derived geometrically with the help of isosceles right 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)

\cos(30°+45° ) = \cos 30°\cos 45° -\sin 30°\sin 45°

= \dfrac{\sqrt{3} }{2}\cdot \dfrac{\sqrt{2} }{2} - \dfrac{1}{2} \cdot \dfrac{\sqrt{2} }{2}

Therefore, rearrange the result and we get the exact value of \cos 75° is as follow

\dfrac{1}{4}(\sqrt{6} -\sqrt{2} )

Using the Microsoft Excel formula

=(SQRT(6)-SQRT(2))/4

to verify it shows the result match the approximate value of \cos 75°, which is 0.258819045

Collected in the board: Trigonometric Values of Special Angles

Steven Zheng posted 5 months ago