Trigonometry is a branch of mathematics that explores the relationships between the angles and lengths of triangles. In this section, we delve into the trigonometric functions: sine (sin), cosine (cos), and tangent (tan), focusing on understanding and sketching their graphs. We will analyse characteristics like amplitude and period, essential for solving various mathematical problems.

Introduction to Trigonometric Functions

Trigonometric functions are fundamental in mathematics, providing a connection between angle measures and ratios of triangle sides. These functions are pivotal in various fields, including physics, engineering, and geometry.

Understanding Sine, Cosine, and Tangent

• Sine (sin x): Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (cos x): Represents the ratio of the adjacent side to the hypotenuse.
• Tangent (tan x): Represents the ratio of the opposite side to the adjacent side.

Graphing Trigonometric Functions

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Sine and Cosine Functions

• General Form: The general form for sine and cosine functions is $y = A \sin(Bx + C) + D$ or $y = A \cos(Bx + C) + D$ where $A$, $B$, $C$, and $D$ are constants.
• Amplitude: The amplitude of a sine or cosine graph is the absolute value of $A$, representing the graph's height from the centre line to its peak or trough.
• Period: The period of a sine or cosine graph is $\dfrac{2\pi}{|B|}$, indicating the length of one complete cycle of the graph.
• Phase Shift: The phase shift, determined by $C$, is the horizontal movement of the graph. A positive $C$ value shifts the graph to the left, while a negative value shifts it to the right.
• Vertical Shift: The vertical shift, determined by $D$, moves the graph up or down.

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Sketching $y = \sin x$ and $y = \cos x$

• y = sin x: Starts at $y = 0$, reaches a maximum of $1$ at $\frac{\pi}{2}$, returns to $0$ at $\pi$, reaches a minimum of $-1$ at $\frac{3\pi}{2}$, and completes one cycle at $2\pi$.

• y = cos x: Starts at $y = 1$, drops to $0$ at $\frac{\pi}{2}$, reaches a minimum of $-1$ at $\pi$, returns to $0$ at $\frac{3\pi}{2}$, and completes one cycle at $2\pi$.

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Tangent Function

• General Form: The general form is $y = A \tan(Bx + C) + D$, with similar constants as sine and cosine but with distinct characteristics.
• Amplitude: The tangent function does not have a maximum amplitude; its values can extend to infinity.
• Period: The period of a tangent graph is $\frac{\pi}{|B|}$, indicating the length of one complete cycle before it repeats.
• Vertical Asymptotes: Unlike sine and cosine, the tangent function has vertical asymptotes where the function is undefined, occurring at $x = \frac{\pi}{2} + n\pi$, where $n$ is an integer.

Sketching $y = \tan x$

The graph of $y = \tan x$ shows a repeating pattern every $\pi$ radians, with vertical asymptotes at $\frac{\pi}{2}$ and $-\frac{\pi}{2}$, where the function approaches infinity.

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Worked Examples

Example 1: Sketching $y = \sin x$

Question: Sketch the graph of $y = \sin x$ over one cycle.

Solution:

• Start by plotting key points at $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$.
• At $x = 0$ and $x = \pi$, $y = 0$. At $x = \frac{\pi}{2}$, $y = 1$, and at $x = \frac{3\pi}{2}$, $y = -1$.
• Connect these points smoothly to complete the sine curve.

Example 2: Sketching $y = \cos x$

Question: Sketch the graph of $y = \cos x$ over one cycle.

Solution:

• Plot key points at $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$.
• At $x = 0$, $y = 1$. At $x = \pi$, $y = -1$. At both $\frac{\pi}{2}$ and $\frac{3\pi}{2}$, $y = 0$.
• Draw a smooth curve to link these points, illustrating the cosine curve.

Example 3: Determining Amplitude and Period

Question: Find the amplitude and period of $y = 3\sin(2x)$.

Solution:

• Amplitude: The amplitude is $|3|$, indicating the graph's peak is 3 units from the centre line.
• Period: The period is $\frac{2\pi}{2} = \pi$, so the graph completes one cycle every $\pi$ radians.

Tips for Success

• Practice sketching these functions by hand to gain a better understanding of their behaviour.
• Remember, the key characteristics of trigonometric graphs include amplitude, period, and vertical shifts.
• Use graph paper and a calculator for accuracy when necessary.