Trigonometric Functions Explained: sin, cos, tan from Unit Circle to Graphs
PUC Mathematics | Trigonometric Functions | Daniel Classes
Why Trig Functions Matter
Sine, cosine, and tangent aren’t just ratios in right triangles. They’re periodic functions that model oscillation — sound waves, light, AC current, radio signals, planetary motion. Understanding them at the function level (not just the triangle level) is essential.
Core shift from PUC to your math life: In school, you memorized sin(30°) = 1/2. In real applications, you need to understand why sin(x) is periodic, how to read its graph, and how to solve equations like sin(x) = 0.7.
The Unit Circle Foundation
Definition: A circle with radius 1 centered at the origin.
For any angle θ (measured counterclockwise from the positive x-axis), the point on the unit circle is (cos θ, sin θ).
Why this matters: This single definition connects the triangle ratios to the periodic functions.
Key angles (in degrees):
- θ = 0°: (cos 0°, sin 0°) = (1, 0)
- θ = 90°: (cos 90°, sin 90°) = (0, 1)
- θ = 180°: (cos 180°, sin 180°) = (-1, 0)
- θ = 270°: (cos 270°, sin 270°) = (0, -1)
- θ = 360°: (cos 360°, sin 360°) = (1, 0) [cycle repeats]
The Three Main Trig Functions
Sine (sin θ)
Definition: The y-coordinate of the point on the unit circle at angle θ.
Domain: All real numbers
Range: [-1, 1]
Period: 2π (or 360°)
Key values:
- sin(0°) = 0
- sin(90°) = 1 [maximum]
- sin(180°) = 0
- sin(270°) = -1 [minimum]
Cosine (cos θ)
Definition: The x-coordinate of the point on the unit circle at angle θ.
Domain: All real numbers
Range: [-1, 1]
Period: 2π (or 360°)
Key values:
- cos(0°) = 1 [maximum]
- cos(90°) = 0
- cos(180°) = -1 [minimum]
- cos(270°) = 0
Tangent (tan θ)
Definition: tan θ = sin θ / cos θ
Domain: All real numbers except where cos θ = 0 (i.e., θ ≠ 90°, 270°, …)
Range: All real numbers
Period: π (or 180°)
Key values:
- tan(0°) = 0
- tan(45°) = 1
- tan(90°) = undefined [vertical asymptote]
Transformations of Trig Functions
General form: f(x) = A sin(B(x – C)) + D
- A = amplitude (stretches/compresses vertically)
- B = frequency (affects period: new period = 2π/B)
- C = phase shift (horizontal translation)
- D = vertical shift
Example: y = 2 sin(2(x – π/4)) + 1
- Amplitude: 2 (oscillates between -2 and 2, then shifted up by 1, so between -1 and 3)
- Period: 2π/2 = π
- Phase shift: π/4 units to the right
- Vertical shift: 1 unit up
How PUC Exams Test Trig Functions
Type 1: Evaluate at Key Angles (2-3 marks)
Example: “Find sin(π/6) + cos(π/3).”
Solution: sin(π/6) = 1/2, cos(π/3) = 1/2, so sum = 1.
Type 2: Find Domain and Range (3-5 marks)
Example: “Find the domain and range of tan(x).”
Solution: Domain: x ≠ π/2 + nπ. Range: all real numbers.
Type 3: Transformations (3-5 marks)
Example: “Describe the graph of y = 2 sin(x – π/4) + 1.”
Solution: Amplitude 2, phase shift π/4 right, vertical shift 1 up.
The Five Common Mistakes
Mistake 1: Confusing sin and cos
Example: “Evaluate sin(90°).”
Wrong: “sin(90°) = 0.” [This is cos(90°)]
Right: “sin(90°) = 1.”
Fix: sin θ = y-coordinate. cos θ = x-coordinate. At 90°, the y-coordinate is 1.
Mistake 2: Forgetting Radians vs Degrees
Example: “Evaluate sin(π/6).”
Wrong: Treats π/6 as degrees instead of radians.
Right: π/6 radians = 30°, so sin(π/6) = 1/2.
Fix: Unless explicitly in degrees, assume radians. 2π radians = 360°.
Mistake 3: Misidentifying the Period
Example: “Find the period of y = sin(2x).”
Wrong: “The period is 2π.” [Ignoring the coefficient]
Right: “The period is 2π/2 = π.”
Fix: For f(x) = sin(Bx), the period is 2π/|B|.
Mistake 4: Confusing Amplitude with Maximum Value
Example: “For y = 3 sin(x) + 2, find the maximum value.”
Wrong: “The maximum is 3.” [This is the amplitude]
Right: “The maximum is 3 + 2 = 5.” [Amplitude plus vertical shift]
Fix: Maximum = amplitude + vertical shift. Minimum = -amplitude + vertical shift.
Mistake 5: Assuming tan(x) Has Range [-1, 1]
Example: “What’s the range of tan(x)?”
Wrong: “It’s [-1, 1].” [This is the range of sin and cos]
Right: “It’s all real numbers.”
Fix: sin and cos are bounded. tan is unbounded.
What to Do Next
To master key values: Memorize sin and cos for 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°.
To master graphs: Sketch y = sin(x), y = cos(x), y = tan(x) and their transformations.
To master transformations: Given a function like y = 2 sin(3(x – π/4)) – 1, identify amplitude, period, phase shift, and vertical shift for 15 examples.
The Problem Vault Alternative
The Problem Vault has 90+ Trigonometric Functions problems from actual PUC exams.