Numeracy

Introduction to Trigonometry




Trigonometry explores the relationships between angles and sides of right-angled triangles, helping calculate unknown sides or angles. It extends beyond triangles into circles and real-world applications like navigation and engineering. Here's a summary:

1. Right-Angled Triangles: Basics

  • Key Elements:
  • The right angle is marked with a small square.
  • ? (theta): The reference angle.
  • Hypotenuse: The triangle's longest side, opposite the right angle.
  • Opposite: The side opposite.
  • Adjacent: The side next to (not the hypotenuse).

2. Pythagoras’ Theorem vs. Trigonometry

  • Pythagoras’ Theorem: For right-angled triangles:
    [
    a^2 + b^2 = c^2
    ]
    Useful for finding one side when the other two are known.
  • Trigonometry: Needed when you know one side and an angle (besides the right angle).

3. Core Trigonometric Functions

Trigonometry is built around three primary functions:
- Sine (sin):
[
\sin(?) = \frac{{Opposite}} / {{Hypotenuse}}
]
- Cosine (cos):
[
\cos(?) = \frac{{Adjacent}} / {{Hypotenuse}}
]
- Tangent (tan):
[
\tan(?) = \frac{{Opposite}} / {{Adjacent}}
]

Mnemonic: Remember SOH CAH TOA for quick recall.
Example: Sin = Opposite / Hypotenuse, Cos = Adjacent / Hypotenuse, Tan = Opposite / Adjacent.

4. Trigonometry and Circles

  • Trigonometric principles extend to all angles (0° to 360°) using unit circles.
  • Quadrants:

  • Values of sine, cosine, and tangent vary depending on the quadrant:

    • Quadrant I: All positive.
    • Quadrant II: Sine positive, others negative.
    • Quadrant III: Tangent positive, others negative.
    • Quadrant IV: Cosine positive, others negative.

  • Unit Circle: A circle with radius 1, used to calculate sin, cos, and tan directly.

5. Trigonometric Graphs

  • Sine and Cosine Graphs:
  • Wavy, cyclic patterns between1 and 1.

  • Period: ( 360° ).

  • Tangent Graph:

  • Repeats every ( 180° ).
  • No maximum or minimum—values go to ±infinity (asymptotes at 90°, 270°, etc.).

6. Inverse Trigonometric Functions

Used to find the angle when a trigonometric ratio is known:
- ( \sin^{-1}(x), \cos^{-1}(x), \tan^{-1}(x) ).
Example:
- If ( \sin(?) = 1 ), then ( \sin^{-1}(1) = 90° ).

7. Other Triangles

Trigonometry applies to non-right triangles via:
- Sine Rule:
[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
]
- Cosine Rule:
[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
]

8. Applications of Trigonometry

  • Navigation: Calculating direction and distance using angles and speed.
  • Engineering: Structural design, satellite positioning, and wave motion.
  • Physics: Harmonic motion (e.g., pendulums), sound, and light waves.

Example: Sailing with Trigonometry

You sail 10 km due east, with a tide pushing 5 km north. What’s your heading?
1. Form a triangle: East = 10 km, North = 5 km.
2. Use tangent:
[
\tan(?) = \frac{{Opposite (5)}} / {{Adjacent (10)}} = 0.5
]
3. Inverse tangent:
[
= \tan^{-1}(0.5) = 26.6°
]
4. Adjust for compass heading:
( 90° - 26.6° = 63.4° ) (between NE and ENE).

Trigonometry is vital for solving real-world problems, from measuring heights to navigating seas!

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