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Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the side lengths and angles of triangles, particularly right-angled triangles. It has wide applications in fields such as science, engineering, astronomy, and video game development. The foundation of trigonometry rests on three primary ratios: sine, cosine, and tangent.

Right-Angled Triangle

Definition Right-Angled Triangle
A right-angled triangle is a triangle with one angle equal to \(90^\circ\). For a given angle \(\theta\) (other than the right angle), we define:
  • Hypotenuse (HYP): The longest side, opposite the right angle.
  • Adjacent Side (ADJ): The side adjacent to the angle \(\theta\), forming one side of the angle.
  • Opposite Side (OPP): The side opposite the angle \(\theta\).
Example
In the triangle below, identify the hypotenuse, the adjacent side, and the opposite side relative to the angle \(\theta\).

  • Hypotenuse: \(\Segment{BC}\)
  • Adjacent side: \(\Segment{AC}\)
  • Opposite side: \(\Segment{AB}\)

Trigonometric Functions


  1. Using a protractor and ruler, draw right-angled triangles with angles \(\theta\) of \(15^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(75^\circ\). Measure the lengths of the hypotenuse and the adjacent side to the nearest millimeter.
  2. Complete the following table with the ratio \(\frac{\text{ADJ}}{\text{HYP}}\) for each angle:
    \(\theta\) \(15^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(75^\circ\)
    \(\dfrac{\text{ADJ}}{\text{HYP}}\)

  1. Examples of right-angled triangles with the specified angles:
  2. The ratio \(\dfrac{\text{ADJ}}{\text{HYP}}\) should yield consistent values for each angle, approximately:
    \(\theta\) \(15^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(75^\circ\)
    \(\dfrac{\text{ADJ}}{\text{HYP}}\) \(\dfrac{4.0}{4.1} \approx 0.98\) \(\dfrac{4.0}{4.6} \approx 0.87\) \(\dfrac{2.5}{3.5} \approx 0.71\) \(\dfrac{2.0}{4.0} = 0.50\) \(\dfrac{2.0}{7.7} \approx 0.26\)


Proposition Trigonometric Ratios
For any two right-angled triangles with the same angle \(\theta\), the ratios \(\dfrac{\text{OPP}}{\text{HYP}}\), \(\dfrac{\text{ADJ}}{\text{HYP}}\), and \(\dfrac{\text{OPP}}{\text{ADJ}}\) are constant.

Consider two right-angled triangles with the same angle \(\theta\). Since their angles are equal, the triangles are similar. Let \(k\) be the scale factor:
The ratios remain constant due to the similarity of the triangles.

Due to the constant nature of the ratio \(\dfrac{\text{ADJ}}{\text{HYP}}\) for any right-angled triangle with angle \(\theta\), we define the cosine function, denoted \(\cos\), where the input is \(\theta\) and the output is \(\dfrac{\text{ADJ}}{\text{HYP}}\). For example:
\(\theta\) \(15^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(75^\circ\)
\(\cos(\theta) = \dfrac{\text{ADJ}}{\text{HYP}}\) \(0.97\) \(0.87\) \(0.71\) \(0.50\) \(0.26\)
For instance, \(\cos(45^\circ) \approx 0.71\).
Definition Trigonometric Functions
In a right-angled triangle with angle \(\theta\):$$\sin(\theta) = \frac{\text{OPP}}{\text{HYP}}, \quad \cos(\theta) = \frac{\text{ADJ}}{\text{HYP}}, \quad \tan(\theta) = \frac{\text{OPP}}{\text{ADJ}}$$
The mnemonic SOH-CAH-TOA helps recall the definitions of sine, cosine, and tangent:
  • Sine = Opposite \(\div\) Hypotenuse
  • Cosine = Adjacent \(\div\) Hypotenuse
  • Tangent = Opposite \(\div\) Adjacent
To aid memorization, you can listen this song: https://www.youtube.com/watch?v=PIWJo5uK3Fo.
Example
In the triangle below, find \(\cos \theta\), \(\sin \theta\), and \(\tan \theta\).

Relative to \(\theta\):
  • Hypotenuse: \(BC = 5\)
  • Adjacent side: \(AB = 4\)
  • Opposite side: \(AC = 3\)
$$\begin{aligned}\cos \theta &= \frac{\text{ADJ}}{\text{HYP}} = \frac{4}{5} \\\sin \theta &= \frac{\text{OPP}}{\text{HYP}} = \frac{3}{5} \\\tan \theta &= \frac{\text{OPP}}{\text{ADJ}} = \frac{3}{4}\end{aligned}$$

Proposition Tangent Formula
$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

$$\begin{aligned}\frac{\sin \theta}{\cos \theta} &= \frac{\dfrac{\text{OPP}}{\text{HYP}}}{\dfrac{\text{ADJ}}{\text{HYP}}} \\&= \frac{\text{OPP}}{\text{HYP}} \times \frac{\text{HYP}}{\text{ADJ}} \\&= \frac{\text{OPP}}{\text{ADJ}} \\&= \tan \theta\end{aligned}$$

Method Using Calculator
Trigonometric ratios for any angle can be calculated using a calculator in degree mode. Ensure your calculator is set to "degrees" before performing calculations.
Example
In the triangle below, find \(x\).

$$\begin{aligned}\cos \theta &= \frac{\text{ADJ}}{\text{HYP}} \\\cos(30^\circ) &= \frac{x}{3} \\x &= 3 \times \cos(30^\circ) \\x &\approx 3 \times 0.866\\x &\approx 2.6 \, \text{cm}\end{aligned}$$

Inverse Trigonometric Functions

Trigonometric ratios can be used to find unknown angles in right-angled triangles when at least two side lengths are known.
Definition Inverse Trigonometric Functions
In a right-angled triangle with angle \(\theta\):$$\theta = \cos^{-1}\left(\frac{\text{ADJ}}{\text{HYP}}\right), \quad \theta = \sin^{-1}\left(\frac{\text{OPP}}{\text{HYP}}\right), \quad \theta = \tan^{-1}\left(\frac{\text{OPP}}{\text{ADJ}}\right)$$
Example
In the triangle below, find \(\theta\).

$$\begin{aligned}\theta &= \cos^{-1}\left(\frac{\text{ADJ}}{\text{HYP}}\right) \\&= \cos^{-1}\left(\frac{0.5}{1}\right) \\&= \cos^{-1}(0.5) \\&= 60^\circ\end{aligned}$$