Trigonometric Ratios and Exact Values

The Fundamental Trigonometric Ratios

Trigonometry in its most basic form is the study of right-angled triangles. For any acute angle $\theta$ in a right-angled triangle, we identify three sides relative to that angle: the hypotenuse (the longest side, opposite the right angle), the opposite (the side across from $\theta$), and the adjacent (the side next to $\theta$ that is not the hypotenuse).

The three primary ratios are defined as follows:

1. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

2. $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

3. $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

These ratios allow us to find an unknown side length if we know one side and one angle, or to find an unknown angle if we know two side lengths. It is sufficient for the TMUA to understand these from the standard definitions of trigonometric functions.

Exact Values for Key Angles

You are expected to know the exact values of the trigonometric functions for specific angles. While these can be memorised, they are easily derived from two standard triangles.

The $45^{\circ}$ Triangle

For an angle of $45^{\circ}$, we use an isosceles right-angled triangle with two sides of length 1. By Pythagoras' theorem, the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$.

From this triangle, we can see:

• $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$
• $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$
• $\tan 45^{\circ} = \frac{1}{1} = 1$

The $30^{\circ}$ and $60^{\circ}$ Triangle

For angles of $30^{\circ}$ and $60^{\circ}$, we consider half of an equilateral triangle with sides of length 2. This creates a right-angled triangle with a hypotenuse of 2 and a base of 1. The vertical height, by Pythagoras, is $\sqrt{2^2 - 1^2} = \sqrt{3}$.

From this triangle, we find:

• $\sin 30^{\circ} = \frac{1}{2}$ and $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$
• $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ and $\cos 60^{\circ} = \frac{1}{2}$
• $\tan 30^{\circ} = \frac{1}{\sqrt{3}}$ and $\tan 60^{\circ} = \sqrt{3}$

For $0^{\circ}$ and $90^{\circ}$, the values are:

• $\sin 0^{\circ} = 0, \cos 0^{\circ} = 1, \tan 0^{\circ} = 0$
• $\sin 90^{\circ} = 1, \cos 90^{\circ} = 0, \tan 90^{\circ}$ is undefined.

Trigonometric Identities

Two essential identities link these ratios. First, the relationship between tangent, sine, and cosine:

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{opposite}/\text{hypotenuse}}{\text{adjacent}/\text{hypotenuse}} = \frac{\sin \theta}{\cos \theta}$

Second, the Pythagorean identity. In a right-angled triangle with hypotenuse 1, the opposite side is $\sin \theta$ and the adjacent side is $\cos \theta$. Applying Pythagoras' theorem gives:

$\sin^2 \theta + \cos^2 \theta = 1$

While we have derived this for acute angles, this formula applies to any angle $\theta$.

Trigonometry as Projection

A useful way to conceptualise sine and cosine is as projection operators. They project a line of length $b$ onto the horizontal $x$ axis or the vertical $y$ axis. In this interpretation, the line segment being projected always has a positive length. The cosine function projects the line onto the $x$ axis, and the sine function projects it onto the $y$ axis.

In the diagrams above, if the angle $\theta$ is obtuse, the projection $b \cos \theta$ becomes negative because it points in the negative $x$ direction. Similarly, the sine projection $b \sin \theta$ becomes negative when the angle points below the $x$ axis.

Application to General and 3D Figures

While you are not required to recall the sine or cosine rules for this specific topic, you must be able to apply trigonometric ratios to general triangles in 2D and 3D by constructing right-angled triangles. This often involves dropping a perpendicular from a vertex to an opposite side (an altitude). In 3D figures, such as pyramids or cuboids, you must identify 2D right-angled triangles within the 3D space, often using the base or the diagonal of a face as one of the sides.