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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Trigonometry is a fundamental branch of mathematics that explores the relationships between the angles and sides of triangles. This week, we will focus on applying trigonometric ratios to solve problems involving right-angled triangles in two dimensions. This is crucial because it allows us to calculate unknown lengths and angles, which has wide-ranging applications in various fields, from surveying land for housing developments in townships to calculating the height of buildings in our cities. Understanding trigonometry is also vital for further studies in mathematics, physics, engineering, and other STEM-related fields.
2.1 Trigonometric Ratios in Right-Angled Triangles A right-angled triangle has one angle equal to 90°. The longest side, opposite the right angle, is called the hypotenuse. The other two sides are named relative to a specific acute angle (an angle less than 90°) we are considering. The side opposite to the angle is the side that does not form one of the arms of the angle. The side adjacent to the angle is the side that forms one of the arms of the angle, but is not the hypotenuse. The three primary trigonometric ratios are defined as follows: Sine (sin): sin(θ) = Opposite / Hypotenuse Cosine (cos): cos(θ) = Adjacent / Hypotenuse Tangent (tan): tan(θ) = Opposite / Adjacent Remember the acronym SOH CAH TOA to easily recall these definitions: Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent 2.2 Special Angles (30°, 45°, 60°) Certain angles appear frequently, and it's useful to know their trigonometric values without relying on a calculator.
These are derived from special triangles: 45-45-90 Triangle: This is an isosceles right-angled triangle. If the two equal sides have length 1, then by the Pythagorean theorem, the hypotenuse has length √
2. Therefore: sin(45°) = 1 / √2 = √2 / 2 (rationalized) cos(45°) = 1 / √2 = √2 / 2 (rationalized) tan(45°) = 1 / 1 = 1 30-60-90 Triangle: This triangle is half of an equilateral triangle. If the side of the equilateral triangle is 2, then the shorter side of the 30-60-90 triangle is 1, and the longer side is √3 (by the Pythagorean theorem).
Therefore: sin(30°) = 1 / 2 cos(30°) = √3 / 2 tan(30°) = 1 / √3 = √3 / 3 (rationalized) sin(60°) = √3 / 2 cos(60°) = 1 / 2 tan(60°) = √3 / 1 = √3 2.3 Solving Right-Angled Triangles To solve a right-angled triangle, you need to find the values of all unknown angles and sides. You can use the trigonometric ratios when you have: One side and one acute angle. Two sides.
Example 1: A ladder leans against a wall, forming an angle of 60° with the ground. The foot of the ladder is 2 meters away from the wall. How high up the wall does the ladder reach?
Solution: Draw a diagram: This helps visualize the problem. Draw a right-angled triangle with the ladder as the hypotenuse, the wall as the opposite side, and the ground as the adjacent side to the 60° angle.
Identify the knowns and unknowns: We know the adjacent side (2m) and the angle (60°). We want to find the opposite side (height up the wall). Choose the appropriate trigonometric ratio: Since we have the adjacent side and want to find the opposite side, we use the tangent ratio: tan(θ) = Opposite / Adjacent Substitute the known values: tan(60°) = Opposite / 2 Solve for the unknown: Opposite = 2 tan(60°) = 2 √3 ≈ 3.46 meters Therefore, the ladder reaches approximately 3.46 meters up the wall.
Example 2: A surveyor needs to determine the height of a cliff. She stands 50 meters away from the base of the cliff and measures the angle of elevation to the top of the cliff as 70°. What is the height of the cliff?
Solution: Draw a diagram: Draw a right-angled triangle with the cliff as the opposite side, the distance to the cliff as the adjacent side, and the line of sight as the hypotenuse.
Identify the knowns and unknowns: We know the adjacent side (50m) and the angle of elevation (70°). We want to find the opposite side (height of the cliff). Choose the appropriate trigonometric ratio: Since we have the adjacent side and want to find the opposite side, we use the tangent ratio: tan(θ) = Opposite / Adjacent Substitute the known values: tan(70°) = Opposite / 50 Solve for the unknown: Opposite = 50 tan(70°) ≈ 50 2.747 ≈ 137.37 meters Therefore, the height of the cliff is approximately 137.37 meters.
Example 3: A ramp is built to access a building. The ramp is 5 meters long and its base is 4 meters from the building. What is the angle of elevation of the ramp?
Solution: Draw a diagram: Draw a right-angled triangle with the ramp as the hypotenuse, the distance from the building as the adjacent side, and the height of the ramp as the opposite side.
Identify the knowns and unknowns: We know the adjacent side (4m) and the hypotenuse (5m). We want to find the angle of elevation. Choose the appropriate trigonometric ratio: Since we have the adjacent and hypotenuse, we use the cosine ratio: cos(θ) = Adjacent / Hypotenuse Substitute the known values: cos(θ) = 4 / 5 = 0.8 Solve for the unknown: θ = cos -1 (0.8) ≈ 36.87° (using the inverse cosine function on a calculator) Therefore, the angle of elevation of the ramp is approximately 36.87°. 2.4 Angles of Elevation and Depression Angle of Elevation: The angle formed between the horizontal line and the line of sight upwards to an object.
Angle of Depression: The angle formed between the horizontal line and the line of sight downwards to an object. These angles are always measured from the horizontal.