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:

1. Use the sine function: $\sin(30°) = \frac{\text{Height of the building}}{100 \text{ m}}$.
2. Knowing $\sin(30°) = \frac{1}{2}$, solve for the height: $\text{Height of the building} = \frac{1}{2} \times 100 \text{ m}$.
3. 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:

1. Use the cosine function: $\cos(60°) = \frac{\text{Base length}}{10 \text{ cm}}.$
2. Knowing $\cos(60°) = \frac{1}{2}$, solve for the base length: $\text{Base length} = \frac{1}{2} \times 10 \text{ cm}$.
3. 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:

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