Fundamental Trigonometric Values and Identities

TL;DR

Trigonometry ties angles to side lengths in right triangles, with sine, cosine, and tangent being the core relationships. You can find exact values for special angles and simplify complex expressions using fundamental identities. Mastering these concepts is crucial for understanding waves, oscillations, and many engineering problems.

1. The Mental Model

Imagine a right-angled triangle. Trigonometry gives you tools to find unknown angles or side lengths if you know some others. It's like a secret decoder ring for triangles (and later, for circles and waves)!

2. The Core Material

Trigonometry is built on the relationships between the angles and sides of a right-angled triangle. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

2.1 SOH CAH TOA: The Basics

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This mnemonic helps you remember the definitions for a right-angled triangle:

• SOH: Sin($\theta$) = Opposite / Hypotenuse
• CAH: Cos($\theta$) = Adjacent / Hypotenuse
• TOA: Tan($\theta$) = Opposite / Adjacent

Here, $\theta$ is one of the acute angles, the hypotenuse is always the longest side (opposite the right angle), the opposite side is across from $\theta$, and the adjacent side is next to $\theta$ (not the hypotenuse).

2.2 Special Angles: Exact Values

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You'll often work with specific angles that have exact, memorable trigonometric values. These are 0°, 30° (or $\pi/6$ rad), 45° (or $\pi/4$ rad), 60° (or $\pi/3$ rad), and 90° (or $\pi/2$ rad).

You can derive these from special triangles: an isosceles right triangle (45-45-90) and an equilateral triangle cut in half (30-60-90).

Remember that tan($\theta$) = sin($\theta$)/cos($\theta$). This is why tan(90°) is undefined (1/0).

2.3 Fundamental Identities

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Identities are equations that are true for all valid values of the variables. These are your building blocks for simplifying and proving more complex trig expressions.

graph TD
    A["Fundamental Trig Identities"] --> B["Reciprocal Identities"]
    A --> C["Quotient Identity"]
    A --> D["Pythagorean Identities"]

    B --> B1["csc(θ) = 1/sin(θ)"]
    B --> B2["sec(θ) = 1/cos(θ)"]
    B --> B3["cot(θ) = 1/tan(θ)"]

    C --> C1["tan(θ) = sin(θ)/cos(θ)"]
    C --> C2["cot(θ) = cos(θ)/sin(θ)"]

    D --> D1["sin²(θ) + cos²(θ) = 1"]
    D --> D2["1 + tan²(θ) = sec²(θ)"]
    D --> D3["1 + cot²(θ) = csc²(θ)"]

    style A fill:#cef,stroke:#333,stroke-width:2px
    style B fill:#dff,stroke:#333,stroke-width:1px
    style C fill:#dff,stroke:#333,stroke-width:1px
    style D fill:#dff,stroke:#333,stroke-width:1px

• Reciprocal Identities: These define the "co-functions" (cosecant, secant, cotangent) by inverting the main ones.
  • csc($\theta$) = 1/sin($\theta$)
  • sec($\theta$) = 1/cos($\theta$)
  • cot($\theta$) = 1/tan($\theta$)
• Quotient Identity: Relates tangent and cotangent to sine and cosine.
  • tan($\theta$) = sin($\theta$)/cos($\theta$)
  • cot($\theta$) = cos($\theta$)/sin($\theta$)
• Pythagorean Identities: These are derived directly from the Pythagorean theorem ($a^2 + b^2 = c^2$) applied to the unit circle (or a right triangle).
  • sin²($\theta$) + cos²($\theta$) = 1 (most important one!)
  • 1 + tan²($\theta$) = sec²($\theta$) (divide the first by cos²($\theta$))
  • 1 + cot²($\theta$) = csc²($\theta$) (divide the first by sin²($\theta$))

3. Worked Example

Simplify the expression: (sin($\theta$) + cos($\theta$))² - 1

Here's how you'd do it step-by-step:

1. Expand the square:
   (sin($\theta$) + cos($\theta$))² = sin²($\theta$) + 2sin($\theta$)cos($\theta$) + cos²($\theta$)

2. Rearrange terms:
   sin²($\theta$) + cos²($\theta$) + 2sin($\theta$)cos($\theta$)

3. Apply Pythagorean Identity (sin²($\theta$) + cos²($\theta$) = 1):
   1 + 2sin($\theta$)cos($\theta$)

4. Substitute back into the original expression:
   (1 + 2sin($\theta$)cos($\theta$)) - 1

5. Simplify:
   2sin($\theta$)cos($\theta$)

So, (sin($\theta$) + cos($\theta$))² - 1 simplifies to 2sin($\theta$)cos($\theta$).

4. Key Takeaways

• SOH CAH TOA is your fundamental guide for relating sides and angles in right triangles.
• Memorize the exact values for 0°, 30°, 45°, 60°, and 90° for quick calculations.
• The Quotient and Reciprocal Identities help you express all trig functions in terms of sine and cosine.
• The Pythagorean Identity (sin²($\theta$) + cos²($\theta$) = 1) is incredibly powerful for simplifying expressions.
• All other Pythagorean identities can be derived from the primary one by dividing by sin²($\theta$) or cos²($\theta$).
• Identities are true for all valid angles, making them useful for simplification and proofs.

Common mistakes to avoid:
- Confusing adjacent/opposite sides when changing the angle of reference.
- Forgetting that sin²($\theta$) is (sin($\theta$))², not sin(sin($\theta$)).
- Trying to "solve" identities; you prove or simplify them, not find a value for $\theta$.
- Incorrectly applying the Pythagorean identity outside of a square relationship (e.g., sin($\theta$) + cos($\theta$) = 1, which is incorrect).

5. Now Try It

Without using a calculator, determine the exact value of (sin(30°) + cos(60°)) / tan(45°). What success looks like: You should arrive at a single numerical value as your answer, showing each step using the special angle values.