IB MYP 4-5 Maths- Graphing trigonometric functions- Study Notes - New Syllabus
IB MYP 4-5 Maths- Graphing trigonometric functions – Study Notes
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- Graphing trigonometric functions
IB MYP 4-5 Maths- Graphing trigonometric functions – Study Notes – All topics
Graphing Trigonometric Functions
Graphing Trigonometric Functions
Trigonometric functions such as sine, cosine, and tangent are periodic functions with characteristic shapes and properties.
1. Sine Function $y = \sin x$
Key Properties:
- Amplitude: 1 (default)
- Period: \( 360^\circ \) or \( 2\pi \) radians
- Domain: All real numbers
- Range: \( -1 \le y \le 1 \)
- Shape: Starts at 0, rises to 1, back to 0, down to -1, and repeats
Graph:
General Form:
\( y = a \sin(bx + c) + d \)
- \( |a| \) = amplitude
- Period = \( \dfrac{360^\circ}{b} \) (or \( \dfrac{2\pi}{b} \) in radians)
- \( c \) = phase shift (horizontal shift)
- \( d \) = vertical shift
Example :
Sketch \( y = 3\sin x \) for \( 0^\circ \le x \le 360^\circ \).
▶️ Answer/Explanation
Amplitude = 3, Period = 360°, no shifts.
Plot key points: (0, 0), (90, 3), (180, 0), (270, -3), (360, 0)
Example :
Sketch \( y = 2\sin(2x) – 1 \) for \( 0^\circ \le x \le 360^\circ \).
▶️ Answer/Explanation
Amplitude = 2, Period = \( \dfrac{360^\circ}{2} = 180^\circ \), Vertical shift = -1.
Key points for one period: (0, -1), (45, 1), (90, -1), (135, -3), (180, -1)
2. Cosine Function $y = \cos x$
Key Properties:
- Amplitude: 1 (default)
- Period: \( 360^\circ \) or \( 2\pi \) radians
- Domain: All real numbers
- Range: \( -1 \le y \le 1 \)
- Shape: Starts at maximum (1), goes to 0, minimum (-1), back to 0, and repeats
Graph Key Points (in degrees):
General Form:
\( y = a \cos(bx + c) + d \)
- \( |a| \) = amplitude
- Period = \( \dfrac{360^\circ}{b} \) (or \( \dfrac{2\pi}{b} \) in radians)
- \( c \) = phase shift (horizontal shift)
- \( d \) = vertical shift
Example :
Sketch \( y = 2\cos x \) for \( 0^\circ \le x \le 360^\circ \).
▶️ Answer/Explanation
Amplitude = 2, Period = 360°, no shifts.
Key points: (0, 2), (90, 0), (180, -2), (270, 0), (360, 1)
Example :
Sketch \( y = -\cos(3x) + 2 \) for \( 0^\circ \le x \le 360^\circ \).
▶️ Answer/Explanation
Amplitude = 1, Reflection in x-axis, Period = \( \dfrac{360^\circ}{3} = 120^\circ \), Vertical shift = +2.
Key points for one cycle: (0, 1), (30, 2), (60, 3), (90, 2), (120, 1)
3. Tangent Function $y = \tan x$
Key Properties:
- Amplitude: None (unbounded)
- Period: \( 180^\circ \) or \( \pi \) radians
- Domain: All real numbers except where \( \cos x = 0 \) (i.e., \( x = 90^\circ, 270^\circ, \dots \))
- Range: \( (-\infty, \infty) \)
- Asymptotes: At \( x = 90^\circ + n \cdot 180^\circ \) (or \( x = \dfrac{\pi}{2} + n\pi \))
- Shape: Passes through origin, increases steeply, with vertical asymptotes
Graph Key Points (in degrees):
General Form:
\( y = a \tan(bx + c) + d \)
- \( |a| \) = vertical stretch
- Period = \( \dfrac{180^\circ}{b} \) (or \( \dfrac{\pi}{b} \) in radians)
- \( c \) = horizontal shift (phase shift)
- \( d \) = vertical shift
Example :
Sketch \( y =4 \tan x \) for \( -180^\circ \le x \le 180^\circ \).
▶️ Answer/Explanation
Period = 180°, Asymptotes at \( x = \pm 90^\circ \), Passes through (0, 0).
Key points: (-90°, undefined), (0°, 0), (45°, 4), (90°, undefined), (135°, -4), (180°, 0).
Example :
Sketch \( y = 2\tan(2x) \) for \( 0^\circ \le x \le 180^\circ \).
▶️ Answer/Explanation
Amplitude = None, Vertical stretch factor = 2, Period = \( \dfrac{180^\circ}{2} = 90^\circ \).
Asymptotes at \( x = 45^\circ, 135^\circ \).
Key points: (0, 0), (22.5°, 2), (45°, undefined), (67.5°, -2), (90°, 0).
Quick Recap Table:
| Function | Period | Range | Domain | Asymptotes |
|---|---|---|---|---|
| \( y = \sin x \) | \( 360^\circ \) / \( 2\pi \) | \([-1, 1]\) | All real numbers | None |
| \( y = \cos x \) | \( 360^\circ \) / \( 2\pi \) | \([-1, 1]\) | All real numbers | None |
| \( y = \tan x \) | \( 180^\circ \) / \( \pi \) | \((-\infty, \infty)\) | All real numbers except \( x = 90^\circ + n \cdot 180^\circ \) | \( x = 90^\circ + n \cdot 180^\circ \), \( n \in \mathbb{Z} \) |
Solving Trigonometric Equations
Solving Trigonometric Equations
Trigonometric equations involve trigonometric functions like \( \sin x, \cos x, \tan x \) and require finding all possible angles within a given interval (commonly \( 0^\circ \leq x < 360^\circ \) or \( 0 \leq x < 2\pi \)).
General Steps:
- Isolate the trigonometric function (e.g., \( \sin x = \text{value} \)).
- Find the reference angle using inverse trig function.
- Determine all solutions in the specified interval based on the quadrant rules.
- Write the general solution if required (using \( +360^\circ k \) or \( +2\pi k \)).
Quadrant Rules:
- \( \sin x > 0 \) in Quadrants I and II.
- \( \cos x > 0 \) in Quadrants I and IV.
- \( \tan x > 0 \) in Quadrants I and III.
Common Solutions:
- \( \sin x = a \) ⇒ \( x = \alpha \) or \( 180^\circ – \alpha \).
- \( \cos x = a \) ⇒ \( x = \alpha \) or \( 360^\circ – \alpha \).
- \( \tan x = a \) ⇒ \( x = \alpha \) or \( 180^\circ + \alpha \).
Example:
Solve \( \sin x = 0.5 \) for \( 0^\circ \le x < 360^\circ \).
▶️ Answer/Explanation
Step 1: Reference angle: \( \sin^{-1}(0.5) = 30^\circ \).
Step 2: \( \sin x \) is positive in Quadrants I and II.
Solutions: \( x = 30^\circ, 180^\circ – 30^\circ = 150^\circ \).
Final Answer: \( x = 30^\circ, 150^\circ \).
Example:
Solve \( 2\cos x – 1 = 0 \) for \( 0^\circ \le x < 360^\circ \).
▶️ Answer/Explanation
Step 1: \( 2\cos x – 1 = 0 \Rightarrow \cos x = 0.5 \).
Step 2: Reference angle: \( \cos^{-1}(0.5) = 60^\circ \).
Step 3: \( \cos x \) is positive in Quadrants I and IV.
Solutions: \( x = 60^\circ, 360^\circ – 60^\circ = 300^\circ \).
Final Answer: \( x = 60^\circ, 300^\circ \).
Example:
Solve \( \tan x = -1 \) for \( 0^\circ \le x < 360^\circ \).
▶️ Answer/Explanation
Step 1: Reference angle: \( \tan^{-1}(1) = 45^\circ \).
Step 2: \( \tan x \) is negative in Quadrants II and IV.
Solutions: \( x = 180^\circ – 45^\circ = 135^\circ, 360^\circ – 45^\circ = 315^\circ \).
Final Answer: \( x = 135^\circ, 315^\circ \).