Trigonometry, an integral part of mathematics, delves into the relationships between angles and sides of triangles. This section elaborates on the exact values of sine, cosine, and tangent for angles of 30°, 45°, and 60°. These angles are crucial due to their regular appearance in mathematical problems and real-world applications.

**Understanding Trigonometric Functions**

Trigonometric functions are fundamental in understanding the dynamics of angles and lengths in a circle, especially the unit circle.

**Sine Function (sin)**

**Definition**: In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse.**Unit Circle Representation**: The y-coordinate of a point on the unit circle.**Significance**: The sine function is pivotal in calculating heights and distances in various fields, including engineering and physics.

**Cosine Function (cos)**

**Definition**: In a right-angled triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.**Unit Circle Representation**: The x-coordinate of a point on the unit circle.**Significance**: Cosine is used extensively in waveform analysis and in determining the phase and amplitude of oscillations.

**Tangent Function (tan)**

**Definition**: In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the adjacent side.**Unit Circle Representation**: The ratio of the y-coordinate to the x-coordinate of a point on the unit circle.**Significance**: Tangent is crucial in surveying, navigation, and in the study of periodic functions.

**Special Angles and Their Trigonometric Values**

Understanding special triangles, such as the 30-60-90 and 45-45-90 triangles, is crucial for comprehending trigonometric ratios.

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**Example 1: Using Sine Function**

Calculate the height of a building given an angle of elevation of 30° from a point 100 meters away.

**Solution:**

- Use the sine function: $\sin(30°) = \frac{\text{Height of the building}}{100 \text{ m}}$.
- Knowing $\sin(30°) = \frac{1}{2}$, solve for the height: $\text{Height of the building} = \frac{1}{2} \times 100 \text{ m}$.
- The height of the building is 50 m.

**Example 2: Using Cosine Function**

Find the length of the base of a right triangle with a hypotenuse of 10 cm and an angle of 60°.

**Solution:**

- Use the cosine function: $\cos(60°) = \frac{\text{Base length}}{10 \text{ cm}}.$
- Knowing $\cos(60°) = \frac{1}{2}$, solve for the base length: $\text{Base length} = \frac{1}{2} \times 10 \text{ cm}$.
- The base length is 5 cm.

**Example 3: Using Tangent Function**

Determine the angle θ made by a 10-meter ladder with the ground, where the foot of the ladder is 6 meters away from the wall.

**Solution:**

- Use the tangent function: $\tan(\theta) = \frac{10 \text{ m}}{6 \text{ m}}$.
- Find the angle using the inverse tangent function: $\theta = \tan^{-1}\left(\frac{10 \text{ m}}{6 \text{ m}}\right)$.
- The angle θ is approximately 59.04°.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.