Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Apply the Pythagorean theorem to find unknown side lengths.
• Define and identify angles of elevation and depression.
• Use sine, cosine, and tangent to solve problems.
• Solve practical problems involving heights and distances.

Lesson summary

This week, we delve into the exciting world of measurement and trigonometry, focusing specifically on applying the Pythagorean theorem and exploring angle of elevation and depression. Measurement and trigonometry are fundamental mathematical tools with wide-ranging applications in everyday life and various professions. From construction and architecture to surveying and navigation, understanding these concepts is crucial. In South Africa, these skills are particularly important for infrastructure development, land management, and various technical fields.

Lesson notes

2.1 The Pythagorean Theorem The Pythagorean theorem is a fundamental relationship in geometry that relates the three sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (called legs or cathetus). Mathematically, this can be expressed as: a² + b² = c² Where: `a` and `b` are the lengths of the legs of the right-angled triangle. `c` is the length of the hypotenuse.

Why it Works: This theorem stems from geometric proofs related to the areas of squares constructed on each side of a right triangle. Numerous proofs exist, but understanding the fundamental relationship between the side lengths is key.

How to Use It: To apply the Pythagorean theorem, follow these steps: Identify the right-angled triangle: Ensure that the triangle in question has a 90-degree angle.

Identify the hypotenuse: The hypotenuse is always the side opposite the right angle.

Identify the legs: The other two sides are the legs.

Apply the formula: Substitute the known values into the equation a² + b² = c².

Solve for the unknown: Use algebraic manipulation to solve for the unknown side length.

Example 1: A ladder is leaning against a wall. The foot of the ladder is 2 meters away from the wall, and the ladder reaches a height of 4 meters on the wall. How long is the ladder?

Solution: This forms a right-angled triangle with the wall and the ground. The ladder is the hypotenuse (c). The distance from the wall (2m) is one leg (a), and the height on the wall (4m) is the other leg (b).

Applying the Pythagorean theorem: a² + b² = c² 2² + 4² = c² 4 + 16 = c² 20 = c² c = √20 ≈ 4.47 meters Therefore, the ladder is approximately 4.47 meters long.

Example 2: Imagine a rectangular field. The length of the field is 12 meters and the width is 5 meters. What is the length of the diagonal of the field?

Solution: The diagonal of a rectangle divides it into two right-angled triangles. The diagonal is the hypotenuse (c). The length (12m) and width (5m) are the legs (a and b).

Applying the Pythagorean theorem: a² + b² = c² 12² + 5² = c² 144 + 25 = c² 169 = c² c = √169 = 13 meters Therefore, the length of the diagonal is 13 meters. 2.2 Angle of Elevation and Angle of Depression These are angles formed between a horizontal line and a line of sight to an object.

Angle of Elevation: The angle formed when an observer looks upwards from the horizontal line to an object above.

Angle of Depression: The angle formed when an observer looks downwards from the horizontal line to an object below. Important

Note: The angle of elevation and the angle of depression are always measured from the horizontal line.

Visual Representation: Imagine a person standing on a cliff looking down at a boat. The angle of depression is the angle between the horizontal line of sight from the person and the line of sight to the boat. Conversely, if a person on the boat looks up at the person on the cliff, the angle of elevation is the angle between the horizontal line of sight from the person on the boat and the line of sight to the person on the cliff.

Example: Imagine a student standing on the top of a school building looking down at the gate. The angle between the horizontal line of sight and the line of sight to the gate is the angle of depression. 2.3 Trigonometric Ratios (Sine, Cosine, Tangent) These ratios relate the angles and sides of a right-angled triangle. They are essential for solving problems involving angles of elevation and depression.

Sine (sin): sin(θ) = Opposite / Hypotenuse Cosine (cos): cos(θ) = Adjacent / Hypotenuse Tangent (tan): tan(θ) = Opposite / Adjacent Where: θ is the angle in question. Opposite is the length of the side opposite the angle θ. Adjacent is the length of the side adjacent to the angle θ (not the hypotenuse). Hypotenuse is the length of the hypotenuse.

Mnemonic: A common mnemonic to remember these ratios is SOH CAH TOA: SOH: Sine = Opposite / Hypotenuse CAH: Cosine = Adjacent / Hypotenuse TOA: Tangent = Opposite / Adjacent Example 3: A flagpole casts a shadow of 10 meters long. The angle of elevation from the tip of the shadow to the top of the flagpole is 60 degrees. How tall is the flagpole?

Solution: This forms a right-angled triangle. We want to find the height of the flagpole (opposite side). We know the length of the shadow (adjacent side) and the angle of elevation.

We can use the tangent ratio: tan(θ) = Opposite / Adjacent tan(60°) = Height / 10 Height = 10 tan(60°) Height ≈ 10 1.732 ≈ 17.32 meters Therefore, the flagpole is approximately 17.32 meters tall. Guided Practice (With Solutions)

Question 1: A bird is sitting on top of a tree. A boy is standing 20 meters away from the base of the tree. The angle of elevation from the boy to the bird is 30 degrees. How tall is the tree?

Solution: We have a right-angled triangle. We need to find the height of the tree (opposite side).