Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply SOH CAH TOA in right-angled triangles.
• Use the Sine Rule to solve triangles.
• Use the Cosine Rule to solve triangles.
• Calculate the area of a triangle with the Area Rule.
• Solve problems with angles of elevation and depression.

Lesson summary

This week, we delve deeper into Trigonometry, focusing on solving two-dimensional (2D) problems using trigonometric ratios, the sine rule, the cosine rule, and the area rule. Trigonometry is not just an abstract concept; it's a powerful tool used in various fields like surveying, navigation, architecture, and even sports analysis. Imagine needing to calculate the height of a cell phone tower in your community without physically climbing it, or determining the angle of elevation to the top of Table Mountain. These are real-world scenarios where trigonometry proves invaluable.

Lesson notes

Trigonometric Ratios (SOH CAH TOA): In a right-angled triangle, the sides are named relative to a specific angle (other than the right angle): Hypotenuse: The longest side, opposite the right angle.

Opposite: The side opposite the angle under consideration.

Adjacent: The side adjacent to (next to) the angle under consideration. The trigonometric ratios are defined as follows: Sine (sin θ) = Opposite / Hypotenuse (SOH) Cosine (cos θ) = Adjacent / Hypotenuse (CAH) Tangent (tan θ) = Opposite / Adjacent (TOA)

Example 1: A ladder 5m long leans against a wall, making an angle of 65° with the ground. How high up the wall does the ladder reach?

Diagram: Draw a right-angled triangle. The ladder is the hypotenuse (5m). The angle between the ladder and the ground is 65°. We need to find the height of the wall, which is the side opposite the 65° angle.

Ratio: Since we have the opposite and hypotenuse, we use the sine function: sin θ = Opposite / Hypotenuse Calculation: sin 65° = Opposite / 5 Opposite = 5 sin 65° Opposite ≈ 5 0.9063 Opposite ≈ 4.53 m Therefore, the ladder reaches approximately 4.53 meters up the wall.

Sine Rule: The Sine Rule relates the sides of a triangle to the sines of their opposite angles: a/sin A = b/sin B = c/sin C This rule is used when you have: Two angles and one side (AAS or ASA) Two sides and a non-included angle (SSA - be aware of the ambiguous case, where two solutions may be possible).

Example 2: In triangle ABC, angle A = 48°, angle B = 75°, and side c = 15 cm. Find the length of side a.

Diagram: Draw a triangle ABC, labeling the given information.

Rule: We use the Sine Rule: a/sin A = c/sin C Calculation: First, find angle C: C = 180° - A - B = 180° - 48° - 75° = 57° a/sin 48° = 15/sin 57° a = (15 sin 48°) / sin 57° a ≈ (15 0.7431) / 0.8387 a ≈ 13.29 cm Therefore, the length of side a is approximately 13.29 cm.

Cosine Rule: The Cosine Rule relates the sides and angles of a triangle: a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B c² = a² + b² - 2ab cos C This rule is used when you have: Three sides (SSS) Two sides and the included angle (SAS)

Example 3: In triangle PQR, p = 8 cm, q = 5 cm, and angle R = 60°. Find the length of side r.

Diagram: Draw triangle PQR, labeling the given information.

Rule: We use the Cosine Rule: r² = p² + q² - 2pq cos R Calculation: r² = 8² + 5² - 2 8 5 cos 60° r² = 64 + 25 - 80 0.5 r² = 89 - 40 r² = 49 r = √49 r = 7 cm Therefore, the length of side r is 7 cm.

Area Rule: The area of a triangle can be calculated using the formula: Area = (1/2)ab sin C = (1/2)bc sin A = (1/2)ac sin B This formula requires two sides and the included angle.

Example 4: In triangle XYZ, x = 10 m, y = 8 m, and angle Z = 120°. Find the area of triangle XY

Z. Diagram: Draw triangle XYZ, labeling the given information.

Rule: We use the Area Rule: Area = (1/2)xy sin Z Calculation: Area = (1/2) 10 8 sin 120° Area = 40 (√3/2) Area ≈ 40 0.866 Area ≈ 34.64 m² Therefore, the area of triangle XYZ is approximately 34.64 m².

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.

Example 5: From the top of a cliff 50m high, the angle of depression to a boat is 35°. How far is the boat from the foot of the cliff?

Diagram: Draw a diagram. The cliff is a vertical line of 50m. The angle of depression from the top of the cliff to the boat is 35°. The distance we need to find is the horizontal distance from the foot of the cliff to the boat.

Analysis: The angle of depression is equal to the angle of elevation from the boat to the top of the cliff (alternate angles). So, we have a right-angled triangle with the opposite side (cliff height) = 50m and the angle = 35°. We need to find the adjacent side (distance to the boat).

Ratio: We use the tangent function: tan θ = Opposite / Adjacent Calculation: tan 35° = 50 / Adjacent Adjacent = 50 / tan 35° Adjacent ≈ 50 / 0.7002 Adjacent ≈ 71.41 m Therefore, the boat is approximately 71.41 meters from the foot of the cliff. Guided Practice (With Solutions)

Question 1: A tower is held by a cable attached to the top of the tower and making an angle of 65° with the ground. If the cable is 30m long, how tall is the tower?

Solution: Diagram: Draw a right-angled triangle. The tower is the vertical side (opposite to the angle). The cable is the hypotenuse (30m). The angle is 65°.

Ratio: We need to find the opposite side, and we have the hypotenuse, so we use the sine function: sin θ = Opposite / Hypotenuse Calculation: sin 65° = Height / 30 Height = 30 sin 65° Height ≈ 30 0.9063 Height ≈ 27.19 m Therefore, the tower is approximately 27.19 meters tall.

Comment: This problem directly applies the sine ratio in a real-world context. We identify the known and unknown quantities and choose the appropriate trigonometric ratio.