Lesson Notes By Weeks and Term v5 - Grade 10
Trigonometry – Week 7 focus
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Trigonometry – Week 7 focus
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Subject: Mathematics
Class: Grade 10
Term: 3rd Term
Week: 7
Theme: General lesson support
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This week, we delve into the fascinating world of Trigonometry. Trigonometry, at its heart, deals with the relationships between the angles and sides of triangles, particularly right-angled triangles. This understanding is fundamental to many fields, from engineering and architecture to navigation and even everyday problem-solving. Imagine needing to calculate the height of a building without physically climbing it, or figuring out the angle of elevation needed to properly install a solar panel for optimal energy capture. Trigonometry provides the tools to tackle these challenges.
Trigonometry focuses on the relationships between the sides and angles of triangles, especially right-angled triangles.
Let's break down the key concepts: 2.
1. Right-Angled Triangle Terminology In a right-angled triangle, one angle is 90 degrees. The side opposite the right angle is called the hypotenuse (the longest side). With respect to any other angle (an acute angle, less than 90 degrees) in the triangle, we define the following: Opposite: The side opposite the chosen angle.
Adjacent: The side next to the chosen angle (not the hypotenuse). It is crucial to remember that "opposite" and "adjacent" are relative to the specific angle you are considering. 2.
2. Trigonometric Ratios The three primary trigonometric ratios relate an angle to the ratios of the sides of a right-angled triangle.
They are: Sine (sin): sin(θ) = Opposite / Hypotenuse Cosine (cos): cos(θ) = Adjacent / Hypotenuse Tangent (tan): tan(θ) = Opposite / Adjacent A helpful mnemonic to remember these is SOH CAH TOA. 2.
3. Using Trigonometric Ratios to Find Sides If you know an angle and the length of one side, you can use trigonometric ratios to find the length of another side.
Example 1: Imagine a telecoms tower casting a shadow. The angle of elevation of the top of the tower from a point 50 meters away from the base of the tower is 60 degrees. Calculate the height of the tower.
Diagram: Draw a right-angled triangle where the tower is the opposite side, the distance from the base is the adjacent side, and the angle of elevation is 60 degrees.
Identify the knowns: Angle (θ) = 60°, Adjacent = 50m, Unknown: Opposite (height of the tower, h)
Choose the appropriate ratio: We have the adjacent and want to find the opposite, so we use tangent (tan = Opposite / Adjacent).
Set up the equation: tan(60°) = h / 50 Solve for h: h = 50 tan(60°)
Calculate: Using a calculator (make sure it's in degree mode!), tan(60°) ≈ 1.
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2. Therefore, h ≈ 50 1.732 ≈ 86.6 meters. The telecoms tower is approximately 86.6 meters high. 2.
4. Using Trigonometric Ratios to Find Angles If you know the lengths of two sides, you can use trigonometric ratios to find the angle. You'll need to use the inverse trigonometric functions (also known as arc functions): Inverse Sine (arcsin or sin⁻¹): If sin(θ) = x, then θ = sin⁻¹(x) Inverse Cosine (arccos or cos⁻¹): If cos(θ) = x, then θ = cos⁻¹(x) Inverse Tangent (arctan or tan⁻¹): If tan(θ) = x, then θ = tan⁻¹(x)
Example 2: A ramp is built to access a building. The ramp is 5 meters long (hypotenuse) and reaches a vertical height of 1 meter (opposite). What is the angle of elevation of the ramp?
Diagram: Draw a right-angled triangle where the ramp is the hypotenuse, the vertical height is the opposite side, and we want to find the angle of elevation (θ).
Identify the knowns: Opposite = 1m, Hypotenuse = 5m, Unknown: Angle (θ)
Choose the appropriate ratio: We have the opposite and the hypotenuse, so we use sine (sin = Opposite / Hypotenuse).
Set up the equation: sin(θ) = 1 / 5 = 0.2 Solve for θ: θ = sin⁻¹(0.2)
Calculate: Using a calculator, sin⁻¹(0.2) ≈ 11.54 degrees. The angle of elevation of the ramp is approximately 11.54 degrees. 2.
5. Angles of Elevation and Depression Angle of Elevation: The angle formed between the horizontal and the line of sight when looking upwards to an object.
Angle of Depression: The angle formed between the horizontal and the line of sight when looking downwards to an object. Remember that the angle of depression from point A to point B is equal to the angle of elevation from point B to point A (alternate angles).
Example 3: From the top of a cliff 50 meters high, the angle of depression to a boat is 35 degrees. How far is the boat from the base of the cliff?
Diagram: Draw a right-angled triangle. The cliff is the opposite side (50m). The angle of depression from the top of the cliff to the boat is 35 degrees. This is equal to the angle of elevation from the boat to the top of the cliff. We want to find the distance from the boat to the base of the cliff, which is the adjacent side.
Identify the knowns: Angle = 35°, Opposite = 50m, Unknown: Adjacent (distance, d)
Choose the appropriate ratio: We have the opposite and want to find the adjacent, so we use tangent (tan = Opposite / Adjacent).
Set up the equation: tan(35°) = 50 / d Solve for d: d = 50 / tan(35°)
Calculate: Using a calculator, tan(35°) ≈ 0.
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0. Therefore, d ≈ 50 / 0.700 ≈ 71.43 meters. The boat is approximately 71.43 meters from the base of the cliff. Guided Practice (With Solutions)
Question 1: A ladder leans against a wall. The ladder is 6 meters long, and the base of the ladder is 2 meters away from the wall. What is the angle the ladder makes with the ground?
Solution: Diagram: Draw a right-angled triangle. The ladder is the hypotenuse (6m), the distance from the wall is the adjacent (2m), and we want to find the angle (θ).