Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. Its origins date back to ancient civilizations, where it was used in astronomy and surveying.

Historical Development

Trigonometry was developed by ancient mathematicians in Greece, India, and the Middle East. Over the centuries, it has evolved to become a crucial part of mathematics with applications across various fields.

Applications of Trigonometry

Trigonometry is widely used in fields such as:

  • Engineering: Designing structures and analyzing mechanical systems.
  • Physics: Describing wave patterns, harmonic motion, and vector analysis.
  • Astronomy: Calculating distances and angles between celestial bodies.

Key Concepts

Understanding Angles

In trigonometry, angles are fundamental. Angles are measured in two units:

  • Degrees (°): There are 360 degrees in a full circle.
  • Radians: The radian is another unit of measurement where a full circle is \(2\pi\) radians.

The Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. Each point on the unit circle represents an angle in radians, and its coordinates relate to the sine and cosine values of that angle.

Unit Circle

Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the lengths of its sides. The main trigonometric ratios are:

  • Sine (sin): Opposite side over Hypotenuse
  • Cosine (cos): Adjacent side over Hypotenuse
  • Tangent (tan): Opposite side over Adjacent side

Practice Questions

  1. Convert 45 degrees to radians.
    Solution

    Using the conversion formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

    So, \( 45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} \) radians.

  2. Find the sine and cosine values for 90 degrees using the unit circle.
    Solution

    At 90 degrees (or \( \frac{\pi}{2} \) radians) on the unit circle, the coordinates are (0, 1).

    Therefore:

    • \( \sin(90^\circ) = 1 \)
    • \( \cos(90^\circ) = 0 \)
  3. Calculate the tangent of 45 degrees.
    Solution

    The tangent ratio is defined as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

    For 45 degrees: \( \sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} \)

    So, \( \tan(45^\circ) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \).

  4. If a point on the unit circle is at (0.6, 0.8), find the angle in degrees and determine the values of sine, cosine, and tangent for that angle.
    Solution

    Using the unit circle, the cosine value is the x-coordinate (0.6) and the sine value is the y-coordinate (0.8).

    The angle with these coordinates is approximately 53.13 degrees.

    • \( \sin(\theta) = 0.8 \)
    • \( \cos(\theta) = 0.6 \)
    • \( \tan(\theta) = \frac{0.8}{0.6} \approx 1.33 \)
→Trigonometric Ratios: Sine, Cosine, and Tangent