Trigonometry: SOHCAHTOA

Understanding Trigonometry

Trigonometry is the study of angles and their relationships with side lengths in right-angled triangles. It helps in calculating unknown sides and angles using three key trigonometric ratios:

• Sine ( $\sin$ )
• Cosine ( $\cos$ )
• Tangent ( $\tan$ )

Each of these ratios is derived from the sides of a right-angled triangle.

 

Labelling a Right-Angled Triangle

Before applying trigonometry, you need to label the sides relative to a given angle $\theta$:

Hypotenuse (H): The longest side, opposite the right angle.

Opposite (O): The side directly opposite the angle $\theta$ .

Adjacent (A): The side next to the angle $\theta$ but not the hypotenuse.

 

 

SOHCAHTOA Mnemonic

SOHCAHTOA is a helpful way to remember the trigonometric ratios:

$$\sin(\theta) = \frac{O}{H} \quad \big(\text{Sine} = \text{Opposite} / \text{Hypotenuse} \big) \\$$

$$\cos(\theta) = \frac{A}{H} \quad \big(\text{Cosine} = \text{Adjacent} / \text{Hypotenuse} \big) \\$$

$$\tan(\theta) = \frac{O}{A} \quad \big(\text{Tangent} = \text{Opposite} / \text{Adjacent} \big) \\$$

 

 

Finding a Missing Side Using SOHCAHTOA

Step-by-Step Approach:

Step 1: Label the triangle’s sides as O, A, H.

Step 2: Identify which trigonometric ratio to use (sin, cos, or tan).

Step 3: Write down the formula and substitute values.

Step 4: Solve for the unknown side by rearranging the equation.

Step 5: Use a calculator to find the value (round if necessary).

 

Example: Finding a Side Length

Given: A right-angled triangle with:

 

 

Step 1: Use  since we have O and H.

$$\sin(35) = \frac{O}{10}$$

 
Step 2: Rearrange the equation:

$$O = 10 \times \sin(35)$$

 
Step 3: Calculate:

$$O \approx 5.74 cm (2 d.p.)$$

 
Final Answer: 5.74 cm

 
Finding a Missing Angle Using SOHCAHTOA

Step-by-Step Approach:

Step 1: Label the triangle’s sides as O, A, H.

Step 2: Identify which trigonometric ratio to use.

Step 3: Write down the formula and substitute values.

Step 4: Use the inverse function ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) to find the angle.

Step 5: Use a calculator and round to 1 decimal place if needed.

 

Example: Finding an Angle

Given: A right-angled triangle with:

 

 

Find the angle $\theta$

Step 1: Use  since we have O and H.

$$\sin(\theta) = \frac{7}{14}$$

 

Step 2: Use the inverse sine function:

$$\theta = \sin^{-1}\bigg(\frac{7}{14}\bigg)$$

 

Step 3: Calculate:

$$\theta \approx \sin^{-1}(0.5) = 30\degree$$

 

Final Answer: $30\degree$

 

 

 

 

Real-World Applications of Trigonometry

Architecture & Engineering – Designing buildings, bridges, and roads.

Aviation & Navigation – Calculating flight paths and ship routes.

Physics & Astronomy – Measuring distances between planets or calculating forces.

Surveying & Cartography – Mapping terrains and land measurements.