Some Special Angles

The multiple and sub-multiple angle formulae in trigonometry are powerful tools for finding the trigonometric ratios of special angles that are not directly available in the standard trigonometric tables. These formulae include the double angle formulas, triple angle formulas, and half-angle formulas, among others. By cleverly applying these, one can derive the sine, cosine, and tangent values for angles like \(15^\circ\), \(18^\circ\), \(22.5^\circ\), and others, which are fractions or multiples of more familiar angles.

For example, to find the sine of \(15^\circ\), one can use the subtraction formula \(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\) with \(a = 45^\circ\) and \(b = 30^\circ\), because the sine and cosine values for \(45^\circ\) and \(30^\circ\) are known. Similarly, the cosine of \(18^\circ\) can be found using the half-angle formula, given that \(36^\circ\) can be related to \(18^\circ\) through the formula \(\cos(2\theta) = 2\cos^2(\theta) - 1\), or its variations. This approach can also be extended to find trigonometric ratios for angles like \(22.5^\circ\) by halving a \(45^\circ\) angle.

Thus, by using these formulae, one can systematically break down less common angles into combinations or halves of standard angles, allowing for the calculation of their trigonometric ratios without direct measurement.

15°

When calculating the trigonometric ratios for a \(15^\circ\) angle using the difference between \(45^\circ\) and \(30^\circ\), we apply specific trigonometric identities. These calculations provide an insightful way to understand the relationships between different angles and their trigonometric ratios.

Sine of \(15^\circ\)

The sine of \(15^\circ\) can be found using the sine difference formula, applying it to the angles \(45^\circ\) and \(30^\circ\):

\[ \sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin(45^\circ)\cos(30^\circ) - \cos(45^\circ)\sin(30^\circ) \]

Given that \(\sin(45^\circ) = \cos(45^\circ) = \frac{1}{\sqrt{2}}\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), and \(\sin(30^\circ) = \frac{1}{2}\), we substitute these values to obtain:

\[ \sin(15^\circ) = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

Cosine of \(15^\circ\)

Similarly, the cosine of \(15^\circ\) is determined by the cosine difference formula:

\[ \cos(15^\circ) = \cos(45^\circ - 30^\circ) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ) \]

Substituting the known values for \(\sin(45^\circ)\) and \(\cos(45^\circ)\):

\[ \cos(15^\circ) = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

Tangent of \(15^\circ\)

The tangent of \(15^\circ\) is the ratio of the sine to the cosine of the angle:

\[ \tan(15^\circ) = \frac{\sin(15^\circ)}{\cos(15^\circ)} = \frac{\frac{\sqrt{3} - 1}{2\sqrt{2}}}{\frac{\sqrt{3} + 1}{2\sqrt{2}}} \]

Rationalizing the denominator, we find:

\[ \tan(15^\circ) = 2 - \sqrt{3} \]

Cotangent of \(15^\circ\)

The cotangent of \(15^\circ\), being the reciprocal of the tangent, is calculated as follows:

\[ \cot(15^\circ) = \frac{1}{\tan(15^\circ)} \]

Given \(\tan(15^\circ) = 2 - \sqrt{3}\), the reciprocal requires rationalization:

\[ \cot(15^\circ) = \frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}} = 2 + \sqrt{3} \]

These formulas and calculations reveal the interconnectedness of trigonometric functions and provide a method for finding the ratios of less common angles such as \(15^\circ\). This approach enhances understanding of trigonometry's fundamental concepts and their practical applications.