Lesson Notes By Weeks and Term v5 - Grade 10
Trigonometry – Week 9 focus
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Trigonometry – Week 9 focus
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Subject: Mathematics
Class: Grade 10
Term: 1st Term
Week: 9
Theme: General lesson support
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This week, we delve deeper into trigonometry, specifically focusing on solving two-dimensional problems using trigonometric ratios (sine, cosine, and tangent) of acute angles. Trigonometry is a powerful tool used in various fields, from surveying and navigation to architecture and engineering. Understanding and applying trigonometric principles allows us to determine unknown lengths and angles in triangles, especially right-angled triangles. This is crucial for solving practical problems related to distances, heights, and angles of elevation and depression. Think about builders ensuring walls are perfectly vertical, or surveyors mapping out land boundaries – they all rely on trigonometry.
a)
Angle of Elevation and Depression: Angle of Elevation: The angle formed between the horizontal line of sight and the line of sight upward to an object. Imagine you are standing on the ground looking up at the top of a building. The angle of elevation is the angle between the ground (your horizontal line of sight) and your line of sight to the top of the building.
Angle of Depression: The angle formed between the horizontal line of sight and the line of sight downward to an object. Imagine you are standing on top of a building looking down at a car on the road. The angle of depression is the angle between your horizontal line of sight and your line of sight to the car. Important
Note: The angle of elevation from point A to point B is equal to the angle of depression from point B to point A (alternate angles, assuming parallel horizontal lines). b)
Trigonometric Ratios (SOH CAH TOA): In a right-angled triangle, the trigonometric ratios relate the angles to the sides.
Sine (sin): sin(θ) = Opposite / Hypotenuse Cosine (cos): cos(θ) = Adjacent / Hypotenuse Tangent (tan): tan(θ) = Opposite / Adjacent Where: θ (theta) is the angle. Opposite is the side opposite to the angle θ. Adjacent is the side adjacent to the angle θ (not the hypotenuse). Hypotenuse is the longest side (opposite the right angle). c)
Solving Problems: Draw a Diagram: Always start by drawing a clear and labeled diagram of the problem. This will help you visualize the situation and identify the relevant sides and angles.
Identify the Knowns and Unknowns: Determine what information is given in the problem (angles, side lengths) and what you need to find. Choose the Appropriate Trigonometric Ratio: Select the trigonometric ratio that relates the known and unknown quantities. Use SOH CAH TOA to help you.
Set up the Equation: Write an equation using the chosen trigonometric ratio.
Solve for the Unknown: Use algebraic manipulation to solve the equation for the unknown quantity.
Include Units: Remember to include the correct units in your answer (e.g., meters, kilometers, degrees).
Check Your Answer: Does your answer seem reasonable in the context of the problem?
Example 1:
A security guard standing 15 meters away from the base of a building observes the top of the building at an angle of elevation of 60°. Calculate the height of the building.
Diagram: Draw a right-angled triangle. The base is 15m (adjacent), the height is the unknown (opposite), and the angle of elevation is 60°.
Knowns: Adjacent = 15m, Angle of elevation = 60°
Unknown: Opposite (height of the building)
Ratio: tan(θ) = Opposite / Adjacent => tan(60°) = Height / 15
Equation: Height = 15 tan(60°)
Solution: Height = 15 1.732 ≈ 25.98 meters
Answer: The height of the building is approximately 25.98 meters.
Example 2:
From the top of a cliff 50 meters high, a fisherman sees a boat at an angle of depression of 35°. How far is the boat from the foot of the cliff?