Comprehensive Study Guide to Right-Angled Triangles and Trigonometry

Pythagoras’ Theorem and Right-Angled Triangles

Pythagoras’ theorem is a fundamental principle in geometry used to determine the lengths of sides within right-angled triangles. The theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship is expressed by the formula:

$h^2 = a^2 + b^2$

In this equation, $h$ represents the length of the hypotenuse, which is the longest side of the triangle and is always located opposite the right angle. The variables $a$ and $b$ represent the lengths of the two shorter sides, often referred to as the legs of the triangle.

Applying Pythagoras’ Theorem

To effectively solve problems using Pythagoras’ theorem, a systematic five-step approach should be followed:

1. Read the question carefully and underline any key terms or quantitative data.
2. Draw a clear diagram of the triangle and label all provided information from the question, including side lengths and the right angle.
3. Decide based on the labeled diagram whether the objective is to determine the length of the hypotenuse ($h$) or the length of one of the shorter sides ($a$ or $b$).
4. Substitute the known values into the theorem $h^2 = a^2 + b^2$ and perform the calculation to find the solution. If finding a shorter side, the formula is rearranged (e.g., $a^2 = h^2 - b^2$).
5. Perform a final check to ensure the answer is reasonable (e.g., the hypotenuse must be the longest side) and verify that the units of measurement are correct.

Trigonometric Ratios

Trigonometry identifies the relationships between the angles and sides of right-angled triangles using three primary ratios: Sine ($\sin$), Cosine ($\cos$), and Tangent ($\tan$). These ratios are defined based on a reference angle, denoted as $\theta$.

The sides of the triangle are identified in relation to $\theta$ as follows:

• The Hypotenuse is the longest side, opposite the right angle.
• The Opposite side is directly across from the angle $\theta$.
• The Adjacent side is next to the angle $\theta$ but is not the hypotenuse.

The ratios are defined as:

• $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ (expressed by the mnemonic SOH)
• $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ (expressed by the mnemonic CAH)
• $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ (expressed by the mnemonic TOA)

Numerical Precision and Calculator Usage

Calculations in trigonometry often involve sub-units of degrees for higher precision. These are categorized into minutes and seconds:

• $1^{\circ} = 60\text{ minutes } (60')$
• $1\text{ minute } (1') = 60\text{ seconds } (60'')$

When finding an unknown side length ($x$):

1. Name the sides of the triangle relative to the given angle (Opposite, Adjacent, or Hypotenuse).
2. Select the appropriate trigonometric ratio ($\sin$, $\cos$, or $\tan$) using the mnemonic SOH CAH TOA based on the known side and the unknown side $x$.
3. Rearrange the equation to isolate the unknown side $x$ as the subject, then use a calculator to evaluate the side length.

When finding an unknown angle ($\theta$):

1. Name the sides of the triangle (Opposite, Adjacent, or Hypotenuse).
2. Use the two given side lengths to determine which trigonometric ratio to use (SOH CAH TOA).
3. Rearrange the ratio to make the angle $\theta$ the subject (often using inverse functions like $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$) and use a calculator to find the value of $\theta$.

Solving Practical Problems

To solve real-world applications of trigonometry, the following steps are used:

1. Read the question thoroughly and underline key terms.
2. Construct a diagram and label it with all information provided.
3. Apply the relevant trigonometric ratios or Pythagoras’ theorem to calculate the solution.

Angles of Elevation and Depression

These angles are measured from a horizontal line of sight:

• Angle of Elevation: The angle measured upwards from the horizontal to an object at a higher level than the observer ($\theta$ is the angle between the horizontal and the line of sight looking up).
• Angle of Depression: The angle measured downwards from the horizontal to an object at a lower level than the observer ($\theta$ is the angle between the horizontal and the line of sight looking down).

Bearings

Bearings are used to communicate direction and are measured in two primary formats:

• Compass Bearing: Directions are given by stating the angle on either side of a North or South baseline. Example: $S60^{\circ}E$ (South 60 degrees East).
• True Bearing: Directions are measured as an angle clockwise from North, ranging from $0^{\circ}$ to $360^{\circ}$. Example: $120^{\circ}$. True bearings are often written as three digits (e.g., $045^{\circ}$ for North-East).

Practice Scenarios and Worked Examples

From the Chapter 4 Review material, several specific scenarios illustrate these principles:

Case Study 1: Finding the Hypotenuse If a triangle has two shorter sides measuring $12\,cm$ and $16\,cm$, the hypotenuse is calculated as: $h = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20\,cm$.

Case Study 2: Calculating Elevation A tower is $80\,m$ tall and an observer is $100\,m$ from the base. To find the angle of elevation ($\theta$): $\tan(\theta) = \frac{80}{100} = 0.8$$\theta = \tan^{-1}(0.8) \approx 38.7^{\circ}$.

Case Study 3: Practical Application (The Cliff and Ship) Susan identifies a ship from the top of a $62\,m$ high cliff with an angle of depression of $31^{\circ}$. The distance from the base of the cliff to the ship can be found using the tangent ratio, where the angle of depression equals the angle of elevation from the ship to the cliff top ($31^{\circ}$). $\tan(31^{\circ}) = \frac{62}{x}$$x = \frac{62}{\tan(31^{\circ})}$ (Value to be calculated correct to one decimal place).

Case Study 4: Navigation (Bearings) Emma travels $8.5\,km$ on a bearing of $N43^{\circ}W$. In this right-angled triangle, the hypotenuse is $8.5\,km$.

• The distance North (adjacent to the $43^{\circ}$ angle) is found using: $x = 8.5 \times \cos(43^{\circ})$.
• The distance West (opposite to the $43^{\circ}$ angle) is found using: $y = 8.5 \times \sin(43^{\circ})$.