Exact Trigonometric Values for Standard Angles

Trigonometric values for standard angles are the foundation of solving geometric and algebraic problems in the ESAT. Rather than relying on a calculator, you should be able to derive these values using two specific triangles and the concept of projections.

Constructing Values for 45 Degrees

To find the values for $45^\circ$, we consider an isosceles right angled triangle. If the two shorter sides are assigned a length of 1, then according to Pythagoras' Theorem, the hypotenuse must be $\sqrt{1^2 + 1^2} = \sqrt{2}$.

From this triangle, we can define the ratios directly:

1. $\sin 45^\circ = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}$
2. $\cos 45^\circ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}$
3. $\tan 45^\circ = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{1}{1} = 1$

Constructing Values for 30 and 60 Degrees

For angles of $30^\circ$ and $60^\circ$, we use an equilateral triangle with side lengths of 2. By bisecting this triangle from one vertex to the midpoint of the opposite side, we create two congruent right angled triangles.

The resulting triangle has a hypotenuse of 2 and a base of 1. The vertical height, calculated using Pythagoras' Theorem, is $\sqrt{2^2 - 1^2} = \sqrt{3}$. The angles in this triangle are $90^\circ$, $60^\circ$, and $30^\circ$.

Using this construction, we can find the exact values for both $30^\circ$ and $60^\circ$:

For $60^\circ$:

1. $\sin 60^\circ = \frac{\sqrt{3}}{2}$
2. $\cos 60^\circ = \frac{1}{2}$
3. $\tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}$

For $30^\circ$:

1. $\sin 30^\circ = \frac{1}{2}$
2. $\cos 30^\circ = \frac{\sqrt{3}}{2}$
3. $\tan 30^\circ = \frac{1}{\sqrt{3}}$

Projections for 0 and 90 Degrees

A useful way to understand trigonometric functions beyond acute angles is to view them as projection operators. In this framework, a line of length $b$ at an angle $\theta$ is projected onto the axes. The cosine function projects the line onto the x axis, while the sine function projects it onto the y axis.

As the angle $\theta$ changes, we can determine the values for $0^\circ$ and $90^\circ$:

1. For $0^\circ$: The line lies entirely on the x axis. The x projection (cosine) is 1 and the y projection (sine) is 0. Therefore, $\sin 0^\circ = 0$, $\cos 0^\circ = 1$, and $\tan 0^\circ = \frac{0}{1} = 0$.
2. For $90^\circ$: The line lies entirely on the y axis. The x projection (cosine) is 0 and the y projection (sine) is 1. Therefore, $\sin 90^\circ = 1$ and $\cos 90^\circ = 0$. Since tangent is defined as $\frac{\sin \theta}{\cos \theta}$, $\tan 90^\circ = \frac{1}{0}$, which is undefined.

Summary of Exact Values

It is essential to learn these values or be able to sketch the triangles quickly. These ratios are consistent whether you use degrees or radians, such as $30^\circ = \frac{\pi}{6}$ or $45^\circ = \frac{\pi}{4}$. You should also be able to identify these values on the standard graphs of sine, cosine, and tangent to solve more complex equations.