Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Trignometry Graphs of Trignometric Ratios
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Trignometry Graphs of Trignometric Ratios
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Subject: General Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 3
Theme: Geometry
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Draw graphs of sineand cosine for angles 0 ≤ x ≤ 360 In terpret/readgraphs of trigonometricratio. Carryout graphicalsolutions of simultaneous linearequation and trigonometric equations.
| :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | | cos x | 1 | 0.87 | 0.5 | 0 | -0.5 | -0.87 | -1 | -0.87 | -0.5 | 0 | 0.5 | 0.87 | 1 | 2.
6. Plotting Graphs
1. Axes: Draw a horizontal x-axis (for angles) and a vertical y-axis (for sine/cosine values) on graph paper.
2. Scale: Choose appropriate scales for both axes. x-axis: Angles typically from 0° to 360°. A common scale is 2 cm to 30° or 2 cm to 45°. y-axis: Values from -1 to
1. A common scale is 2 cm to 0.2 units or 2 cm to 0.5 units. Ensure -1 and 1 are clearly marked.
3. Labeling: Label the x-axis as "Angle (x°)" and the y-axis as "y" or "sin x / cos x". Clearly mark the origin (0,0).
4. Plotting Points: Carefully plot each (x, y) coordinate pair from the table.
5. Drawing the Curve: Join the plotted points with a smooth, continuous curve. Avoid drawing straight lines between points. The curve should be freehand and flow smoothly through the points. 2.
7. Interpreting/Reading Graphs of Trigonometric Ratios Once a graph is drawn, students can: Read values: For a given angle `x`, find the corresponding `y` value (e.g., find sin 150° from the graph).
Find angles: For a given `y` value, find the corresponding angle(s) `x` within the specified range (e.g., find `x` when sin x = 0.5).
Identify Maximum/Minimum: Determine the highest and lowest points on the graph and their corresponding angle values.
Determine Period: Observe the interval over which the graph completes one full cycle.
Example 3: Interpretation of y = sin x graph To find `sin 150°`: Locate `150°` on the x-axis, move vertically to the curve, then horizontally to the y-axis. The reading should be approximately 0.
5. To find `x` when `sin x = 0.7`: Locate `0.7` on the y-axis, move horizontally to intersect the curve, then vertically down to the x-axis. There will be two solutions (e.g., approx. 44° and 136°). 2.
8. Graphical Solutions of Simultaneous Equations This involves plotting two graphs on the same set of axes and finding their point(s) of intersection. 2.8.
1. Solving Trigonometric Equations (e.g., sin x = k) Draw the graph of `y = sin x`. Draw the horizontal line `y = k` (where `k` is a constant between -1 and 1). The x-coordinates of the intersection points are the solutions to the equation.
Example 4: Solve sin x = 0.5 graphically for 0° ≤ x ≤ 360°
1. Draw `y = sin x` (as in Example 1).
2. Draw the line `y = 0.5`.
3. Identify the x-coordinates where the line `y = 0.5` intersects the curve `y = sin x`. The solutions should be approximately 30° and 150°. 2.8.
2. Solving Simultaneous Linear and Trigonometric Equations (e.g., y = sin x and y = mx + c) Rearrange the equation into two functions if necessary (e.g., `sin x = x/180` becomes `y = sin x` and `y = x/180`). Draw the graph of the trigonometric function (e.g., `y = sin x`). Draw the graph of the linear function (e.g., `y = x/180`) on the same axes. A table of values is also needed for the linear function (e.g., for `y = x/180`, if x=0, y=0; if x=180, y=1; if x=360, y=2). The x-coordinates of the intersection points are the solutions to the combined equation.
Example 5: Solve sin x = x/180 graphically for 0° ≤ x ≤ 360°
1. Draw `y = sin x`.
2. Draw `y = x/180`.
For this line: If `x = 0`, `y = 0/180 = 0`. If `x = 180`, `y = 180/180 = 1`. If `x = 360`, `y = 360/180 = 2`. * Plot (0,0), (180,1), (360,2) and draw a straight line through them.
3. Observe the intersection points. There will be one at `x = 0°` and another point around `x = 180°`. (
Note: for `x=180`, sin 180 = 0, but x/180 =
1. So they x ≤ 360°
1. Draw `y = sin x`.
2. Draw `y = x/180`.
For this line: If `x = 0`, `y = 0/180 = 0`. If `x = 180`, `y = 180/180 = 1`. If `x = 360`, `y = 360/180 = 2`. Plot (0,0), (180,1), (360,2) and draw a straight line through them.
3. Observe the intersection points. There will be one at `x = 0°` and another point around `x = 180°`. (
Note: for `x=180`, sin 180 = 0, but x/180 =
1. So they don't meet at
1
8
0. They meet closer to 190-200 if one was to extend the y-axis range).
Let's recheck the values: - At x=0, sin 0 = 0, x/180 =
0. So (0,0) is a solution. - The line goes from (0,0) to (180,1). The sine graph goes from (0,0) to (90,1) and then (180,0). So the line y=x/180 intersects y=sin x between 0 and 90, and then again between 180 and 270 (if the y-axis scale can accommodate y values up to 2). A better example might be needed or adjust range to 0 to
1
8
0. Let's refine Example 5 to make it more suitable for standard graph scale: Refined Example 5: Solve `sin x = x/360` graphically for 0° ≤ x ≤ 360°**
1. Draw `y = sin x`.
2. Draw `y = x/360`.
For this line: If `x = 0`, `y = 0`. If `x = 180`, `y = 0.5`. * If `x = 360`, `y = 1`.
3. Plot (0,0), (180,0.5), (360,1) and draw a straight line.
4. The intersection points will be at `x = 0°` and another solution around `x = 170°` (where the line intersects the curve `y = sin x` near its peak and descending arm). This section outlines the fundamental concepts required to understand and construct trigonometric graphs. 2.
1. Introduction to Trigonometric Graphs Trigonometric graphs are visual representations of how trigonometric ratios (sine, cosine, tangent) vary with changes in angle. Unlike linear graphs, trigonometric graphs are typically periodic, meaning their pattern repeats at regular intervals. For this lesson, the focus is on sine and cosine graphs for angles between 0° and 360°. 2.
2. The Unit Circle and Trigonometric Values The unit circle (a circle with radius 1 unit centered at the origin) is the foundation for understanding trigonometric values for angles beyond 90°. For any angle `x` (measured counter-clockwise from the positive x-axis), the coordinates `(P(x,y))` of the point where the terminal arm of the angle intersects the unit circle define: `cos x = x-coordinate` `sin x = y-coordinate` This relationship helps to visualize how `sin x` and `cos x` vary between -1 and 1 as `x` rotates from 0° to 360°. 2.
3. Properties of y = sin x (for 0° ≤ x ≤ 360°)
Shape: A continuous, wave-like curve.
Range: The y-values range from -1 to
1. The maximum value is 1, and the minimum value is -
1. Period: The pattern of the graph repeats every 360°.
Key Points: `sin 0° = 0` `sin 90° = 1` (Maximum) `sin 180° = 0` `sin 270° = -1` (Minimum) `sin 360° = 0` Symmetry: The graph is symmetrical about x = 90° and x = 270°. 2.
4. Properties of y = cos x (for 0° ≤ x ≤ 360°)
Shape: A continuous, wave-like curve, similar to sine but shifted.
Range: The y-values range from -1 to
1. The maximum value is 1, and the minimum value is -
1. Period: The pattern of the graph repeats every 360°.
Key Points: `cos 0° = 1` (Maximum) `cos 90° = 0` `cos 180° = -1` (Minimum) `cos 270° = 0` `cos 360° = 1` Symmetry: The graph is symmetrical about x = 0° (or 360°) and x = 180°. 2.
5. Constructing Tables of Values To draw a smooth trigonometric graph, a table of values is required.
Intervals: Choose appropriate intervals for `x` (e.g., 30°, 45°, or 60°). Smaller intervals result in more plotted points and a smoother curve. For SS3, 30° intervals are generally sufficient.
Calculation: Use a scientific calculator (ensuring it is in DEGREE mode) to find the sine or cosine of each angle. Round values to 2 or 3 decimal places for plotting accuracy.
Example 1: Constructing a table for y = sin x (0° ≤ x ≤ 360° at 30° intervals) | x (degrees) | 0° | 30° | 60° | 90° | 120° | 150° | 180° | 210° | 240° | 270° | 300° | 330° | 360° | | :---------- | :-- | :--- | :--- | :--- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | | sin x | 0 | 0.5 | 0.87 | 1 | 0.87 | 0.5 | 0 | -0.5 | -0.87 | -1 | -0.87 | -0.5 | 0 | Example 2: Constructing a table for y = cos x (0° ≤ x ≤ 360° at 30° intervals) | x (degrees) | 0° | 30° | 60° | 90° | 120° | 150° | 180° | 210° | 240° | 270° | 300° | 330° | 360° | | :---------- | :-- | :--- | :--- | :--- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | :---- | | cos x | 1 | 0.87 | 0.5 | 0 | -0.5 | -0.87 | -1 | -0.87 | -0.5 | 0 | 0.5 | 0.87 | 1 | 2.
6. Plotting Graphs
1. Axes: Draw a horizontal x-axis (for angles) and a vertical y-axis (for sine/cosine values) on graph paper.
2. Scale: Choose appropriate scales for both axes. * x-axis: Angles typically from 0° to 360°. A common scale is 2 cm to 30° This section provides a structured approach for lesson delivery. 3.
1. Introduction (10 minutes)
Teacher Activity: Initiate a brief review of basic trigonometric ratios (SOH CAH TOA) and their values for special angles (0°, 30°, 45°, 60°, 90°). Briefly introduce the concept of the unit circle and how `sin x` and `cos x` values are derived from x and y coordinates, emphasizing values beyond 90° (e.g., sin 120°, cos 210°).
Pose a question: "How can we visually represent how these values change continuously as the angle increases?" Student Activity: Respond to review questions and share their understanding of trigonometric ratios and the unit circle. Engage in a short discussion about continuous change and graphical representation. 3.
2. Exploring Properties and Table Construction (20 minutes)
Teacher Activity: Explicitly state the properties of `y = sin x` and `y = cos x` (range, period, key points 0°, 90°, 180°, 270°, 360°). Demonstrate step-by-step how to construct a table of values for `y = sin x` for `0° ≤ x ≤ 360°` using 30° intervals. Emphasize using a scientific calculator in DEGREE mode and rounding to 2 decimal places. Guide students in selecting an appropriate scale for axes on graph paper, highlighting the typical range for x and y values.
Student Activity: Copy the tables and properties into their notebooks. Practise using their calculators to find sine values for various angles. Participate in a guided session to construct a table of values for `y = cos x` (0° ≤ x ≤ 360°). 3.
3. Graph Plotting and Interpretation (30 minutes)
Teacher Activity: Distribute graph papers to students. On a large chalkboard graph or projector, demonstrate how to plot points from the `y = sin x` table. Emphasize drawing a smooth curve freehand, connecting the points without sharp corners. Guide students on labelling axes correctly and clearly indicating the chosen scale.
Demonstrate how to interpret the graph: reading `y` for a given `x`, finding `x` for a given `y`, and identifying maximum/minimum values. Supervise students as they plot their own `y = sin x` graph.
Student Activity: Plot the graph of `y = sin x` on their graph papers, following the teacher's demonstration and guidance. Practise interpreting their `y = sin x` graph by answering questions like: "What is sin 240°?", "For what value(s) of x is sin x = 0.8?". Commence plotting the graph of `y = cos x` on the same or a new set of axes. 3.
4. Graphical Solutions of Equations (25 minutes)
Teacher Activity: Explain the concept of solving equations graphically by finding intersection points of two graphs. Demonstrate solving a trigonometric equation, e.g., `sin x = 0.7`, using the already drawn `y = sin x` graph by drawing a horizontal line `y = 0.7`. Introduce solving simultaneous linear and trigonometric equations, e.g., `sin x = x/180`. Show how to rearrange into two functions (`y = sin x` and `y = x/180`) and then plot the linear function on the same axes. Emphasize using specific points to plot the straight line (e.g., for `y = x/180`, plot `(0,0)`, `(180,1)`). Guide students to identify and read the x-coordinates of intersection points.
Student Activity: On their graph paper, use their `y = sin x` graph to solve a given trigonometric equation (e.g., `sin x = -0.5`). Work in pairs or small groups to plot a linear graph (e.g., `y = x/360`) on the same axes as their `y = sin x` graph and find the intersection points, solving a simultaneous equation. Discuss the accuracy of their graphical solutions and potential challenges. 3.
5. Wrap-up and Assignment (5 minutes)
Teacher Activity: Summarise key learning points: constructing tables, plotting smooth curves, interpreting graphs, and solving equations graphically. Assign independent practice questions.
Student Activity: Ask clarifying questions. Note down homework assignment.
Electrical Power Generation and Distribution: In Nigeria, the electricity grid primarily relies on alternating current (AC). The voltage and current in AC circuits vary sinusoidally (following sine or cosine graphs) over time. Electrical engineers at power plants like Egbin or hydropower stations like Kainji use their understanding of these graphs to design generators, transformers, and transmission lines, ensuring stable power delivery to homes and industries. Analyzing phase shifts, amplitude, and frequency in these graphs is critical for grid stability.
Telecommunications and Broadcast: Radio, television, and mobile phone signals in Nigeria are transmitted as electromagnetic waves, which exhibit wave-like behavior modeled by trigonometric functions. For instance, engineers at telecommunication companies (e.g., MTN, Glo, Airtel) or broadcast stations (e.g., NTA, Channels TV) use these graphs to understand signal propagation, design antennas, and ensure clear communication across varying terrains, from the bustling streets of Lagos to rural communities in Sokoto or Yenagoa.
Tide Prediction in Coastal Areas: Nigeria has an extensive coastline along the Atlantic Ocean, vital for maritime activities and fishing. The rise and fall of ocean tides are periodic phenomena that can be accurately modeled using sine and cosine functions. Naval officers, fishermen, and port authorities in cities like Calabar, Port Harcourt, and Lagos utilize tide tables (which are generated from trigonometric models) to predict high and low tides, crucial for safe navigation, docking large vessels, and optimizing fishing schedules.