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Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the side lengths and angles of triangles, especially right-angled triangles. It is widely used in science, engineering, astronomy, and even video game development. The foundation of trigonometry lies in three main 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 next to the angle \(\theta\), which is not the hypotenuse.
  • Opposite Side (OPP): The side facing the angle \(\theta\).
Example
In the triangle below, identify the hypotenuse, the adjacent side, and the opposite side relative to 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 one angle \(\theta\) equal to \(15^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), or \(75^\circ\). 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. Here are examples of right-angled triangles for the specified angles. In each triangle, the side adjacent to \(\theta\) (ADJ) and the hypotenuse (HYP) are labeled so you can measure and verify the ratios.
  2. The ratio \(\dfrac{\text{ADJ}}{\text{HYP}}\) for each angle, using a calculator:
    \(\theta\) \(15^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(75^\circ\)
    \(\dfrac{\text{ADJ}}{\text{HYP}}\) \(0.97\) \(0.87\) \(0.71\) \(0.50\) \(0.26\)


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. Therefore, the ratios are constant for a given angle.

This means that for any right-angled triangle with the same angle \(\theta\), the ratio \(\dfrac{\text{ADJ}}{\text{HYP}}\) is always the same. This is why we define the cosine function, denoted \(\cos\), such that:$$\cos(\theta) = \frac{\text{ADJ}}{\text{HYP}}$$For example:
\(\theta\) \(15^\circ\) \(30^\circ\) \(45^\circ\) \(60^\circ\) \(75^\circ\)
\(\cos(\theta)\) \(0.97\) \(0.87\) \(0.71\) \(0.50\) \(0.26\)
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 remember the definitions:
  • Sine = Opposite \(\div\) Hypotenuse
  • Cosine = Adjacent \(\div\) Hypotenuse
  • Tangent = Opposite \(\div\) Adjacent
To help you memorize, listen to 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
$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

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

Method Using Calculator
Trigonometric ratios for any angle can be calculated using a calculator in degree mode. Make sure your calculator is set to "degrees" before calculating.
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 an 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 the angle \(\theta\).

We know the lengths of the adjacent side (\(AB = 0.5\)) and the hypotenuse (\(BC = 1\)) relative to \(\theta\). We can use the inverse cosine function:$$\begin{aligned}\theta &= \cos^{-1}\left(\frac{\text{ADJ}}{\text{HYP}}\right) \\ &= \cos^{-1}\left(\frac{0.5}{1}\right) \\ &= 60^\circ\end{aligned}$$

Solving Real-World Trigonometry Problems

Trigonometric ratios are powerful tools for solving a wide range of problems involving right-angled triangles, especially in real-world contexts.To solve these problems effectively, follow the structured steps below:
Method Solving Real-World Trigonometry Problems
  • Draw a clear diagram representing the situation described in the problem.
  • Label the unknown (side or angle) you need to find. Use \(x\) for a side and \(\theta\) for an angle if possible.
  • Identify a right-angled triangle within your diagram.
  • Write an equation relating an angle and two sides of the triangle using the appropriate trigonometric ratio.
  • Solve the equation to find the unknown value.
  • State your answer clearly, including appropriate units, in a complete sentence.