Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Trigonometry function
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Trigonometry function
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Subject: Further Mathematics
Class: Senior Secondary 2
Term: 1st Term
Week: 1
Theme: Pure Mathematics
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Know the six important trignometric functions of angles of any magnitude Identify range and dimension of trignometric ratios Drawing of graphs of trignometric ratios Identifying relationships between graphs of trig ratios e.g sin x and sin 2x graphs In verse trignometric ratios Solution of equation of trignometric ratios Proof of simple trignometric identities
Trigonometric functions are defined using the coordinates of a point on the terminal arm of an angle in a unit circle, or more generally, a circle of radius r centered at the origin (0,0). Consider a point P(x, y) on the circumference of a circle with radius r. Let $\theta$ be the angle formed by the positive x-axis and the line segment O
P. The six trigonometric functions are: Sine (sin $\theta$): $\frac{opposite}{hypotenuse} = \frac{y}{r}$ Cosine (cos $\theta$): $\frac{adjacent}{hypotenuse} = \frac{x}{r}$ Tangent (tan $\theta$): $\frac{opposite}{adjacent} = \frac{y}{x}$ (where $x \neq 0$)
Their reciprocal functions are: Cosecant (cosec $\theta$ or csc $\theta$): $\frac{1}{\sin \theta} = \frac{r}{y}$ (where $y \neq 0$) Secant (sec $\theta$): $\frac{1}{\cos \theta} = \frac{r}{x}$ (where $x \neq 0$) Cotangent (cot $\theta$): $\frac{1}{\tan \theta} = \frac{x}{y}$ (where $y \neq 0$)
Angles of Any Magnitude (CAST Rule): The signs of the trigonometric functions depend on the quadrant in which the terminal arm of the angle $\theta$ lies. Quadrant I (0° to 90°): All functions are positive. Quadrant II (90° to 180°): Sine and its reciprocal (cosecant) are positive. Quadrant III (180° to 270°): Tangent and its reciprocal (cotangent) are positive. Quadrant IV (270° to 360°): Cosine and its reciprocal (secant) are positive. This is commonly remembered by the acronym "CAST" (starting from Q4 and moving anti-clockwise), or "All Students Take Chemistry" (starting from Q1 and moving anti-clockwise).
Reference Angle: For any angle $\theta$, its reference angle ($\alpha$) is the acute angle formed by the terminal arm and the x-axis. Q1: $\alpha = \theta$ Q2: $\alpha = 180° - \theta$ or $\pi - \theta$ Q3: $\alpha = \theta - 180°$ or $\theta - \pi$ Q4: $\alpha = 360° - \theta$ or $2\pi - \theta$ Worked Example 2.1.1: Find the exact values of $\sin 210°$, $\cos 300°$, and $\tan 135°$.
Solution: $\sin 210°$: $210°$ is in Quadrant III. In Q3, sine is negative. Reference angle $\alpha = 210° - 180° = 30°$. $\sin 210° = -\sin 30° = -\frac{1}{2}$. $\cos 300°$: $300°$ is in Quadrant IV. In Q4, cosine is positive. Reference angle $\alpha = 360° - 300° = 60°$. $\cos 300° = \cos 60° = \frac{1}{2}$. $\tan 135°$: $135°$ is in Quadrant II. In Q2, tangent is negative. Reference angle $\alpha = 180° - 135° = 45°$. $\tan 135° = -\tan 45° = -1$.
Domain: The set of all possible input values (angles) for which the function is defined.
Range: The set of all possible output values that the function can produce. | Function | Domain | Range | | :------------- | :----------------------------------- | :----------------------------------- | | sin x | All real numbers (R) or $(-\infty, \infty)$ | $[-1, 1]$ | | cos x | All real numbers (R) or $(-\infty, \infty)$ | $[-1, 1]$ | | tan x | $x \neq (2n+1)\frac{\pi}{2}$, $n \in Z$ ($x \neq 90°, 270°, ...$) | All real numbers (R) or $(-\infty, \infty)$ | | cosec x | $x \neq n\pi$, $n \in Z$ ($x \neq 0°, 180°, 360°, ...$) | $(-\infty, -1] \cup [1, \infty)$ | | sec x | $x \neq (2n+1)\frac{\pi}{2}$, $n \in Z$ ($x \neq 90°, 270°, ...$) | $(-\infty, -1] \cup [1, \infty)$ | | cot x | $x \neq n\pi$, $n \in Z$ ($x \neq 0°, 180°, 360°, ...$) | All real numbers (R) or $(-\infty, \infty)$ | Trigonometric graphs are periodic, meaning they repeat their pattern over a regular interval called the period. Period of sin x, cos x, cosec x, sec x: $360°$ or $2\pi$ radians. Period of tan x, cot x: $180°$ or $\pi$ radians.
Amplitude: For sin x and cos x, the amplitude is
1. It is half the difference between the maximum and minimum values.
Steps for drawing graphs: Create a table of values for x (angle) and y (function value) at key points (0°, 90°, 180°, 270°, 360° or 0, $\pi/2$, $\pi$, $3\pi/2$, $2\pi$). Plot the points on a coordinate plane. Connect the points with a smooth curve, respecting the periodic nature.
Graphs to be drawn and understood: $y = \sin x$: Starts at (0,0), rises to 1 at 90°, back to 0 at 180°, down to -1 at 270°, back to 0 at 360°. $y = \cos x$: Starts at (0,1), decreases to 0 at 90°, to -1 at 180°, to 0 at 270°, back to 1 at 360°. $y = \tan x$: Undefined at 90°, 270°, etc. (vertical asymptotes). Passes through (0,0), increases from $-\infty$ to $\infty$ in intervals like $(-90°, 90°)$. (Teacher should sketch these on a whiteboard during the lesson). Understanding transformations of trigonometric graphs is crucial. For a general trigonometric function $y = A \sin(Bx + C) + D$ or $y = A \cos(Bx + C) + D$: A (Amplitude): Determines the vertical stretch or compression. The amplitude is $|A|$.
B (Period): Affects the horizontal stretch or compression. The period is $\frac{360°}{|B|}$ or $\frac{2\pi}{|B|}$ for sine/cosine/secant/cosecant. For tangent/cotangent, it's $\frac{180°}{|B|}$ or $\frac{\pi}{|B|}$.
C (Phase Shift): Determines the horizontal shift. A positive C shifts the graph to the left, negative to the right. Phase shift = $-\frac{C}{B}$.
D (Vertical Shift): Determines the vertical shift. Positive D shifts graph upwards, negative D shifts downwards. Relationship between $y = \sin x$ and $y = \sin 2x$: $y = \sin x$: Amplitude = 1, Period = $360°$. $y = \sin 2x$: Amplitude =
1. Period = $\frac{360°}{2} = 180°$. The graph of $y = \sin 2x$ is a horizontal compression of the graph of $y = \sin x$ by a factor of
2. It completes two full cycles in the same interval that $y = \sin x$ completes one cycle. Worked Example 2.4.1: Describe the relationship between the graph of $y = \cos x$ and $y = 3 \cos x$.
Solution: The graph of $y = 3 \cos x$ has an amplitude of 3, while $y = \cos x$ has an amplitude of
1. Both have a period of $360°$. The graph of $y = 3 \cos x$ is a vertical stretch of the graph of $y = \cos x$ by a factor of
3. Its maximum value is 3 and minimum value is -3.
Surveying and Land Management (Construction & Urban Planning): Trigonometry is indispensable in surveying. Surveyors in Nigeria use trigonometric functions (especially sine, cosine, and tangent) to calculate distances, angles, and elevations of land.
This is crucial for: Mapping out new roads and bridges: Determining gradients and lengths.
Designing building foundations: Ensuring structures are level and stable.
Boundary demarcation: Establishing property lines in communities, often using instruments like theodolites which rely on angular measurements.
Infrastructure development: Planning for pipelines, power lines, and communication masts across varying terrain. Architecture and Engineering (Building & Structural Design): Architects and engineers in Nigeria apply trigonometric principles in designing aesthetically pleasing and structurally sound buildings, from residential homes to multi-story office complexes in cities like Lagos or Abuja.
Roof pitches: Calculating the angle of a roof for proper water runoff, which is vital in Nigeria's rainy seasons.
Bridge construction: Analyzing forces and stresses on beams and supports, ensuring stability against strong winds or heavy loads.
Ramps and access ways: Designing wheelchair ramps with appropriate inclinations for accessibility, using tangent functions. Navigation and Communication (Transportation & Telecommunications): Maritime and Aviation: Pilots and ship captains in Nigeria use trigonometry for navigation, calculating their bearing, distance to destinations, and altitude/depth. For example, a ship captain might use the angle of elevation of a lighthouse to determine distance from the shore.
Telecommunications: The principles of wave functions, which are based on sine and cosine graphs, are fundamental to designing and optimizing communication networks, including radio signals, television broadcasts, and mobile phone towers (like those of MTN, Glo, Airtel). Engineers analyze the amplitude, frequency, and phase of these waves to ensure clear signal transmission across different terrains in Nigeria.