Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply the Sine Rule to find unknown sides and angles.
• Apply the Cosine Rule to find unknown sides and angles.
• Calculate the area of a triangle using the Area Rule.
• Choose the appropriate rule for a given problem.
• Solve real-world problems using these rules.

Lesson summary

Trigonometry is a vital branch of mathematics that deals with the relationships between the sides and angles of triangles. This week, we delve into the Sine Rule, Cosine Rule, and Area Rule, which are essential tools for solving problems involving non-right-angled triangles. These rules are widely used in various fields, from surveying land in our communities to calculating distances in navigation and construction projects that build our infrastructure. Understanding these rules allows us to analyze and solve real-world problems that involve angles and distances.

Lesson notes

2. 1. Sine Rule The Sine Rule states that in any triangle ABC, the ratio of the length of a side to the sine of the angle opposite that side is constant. Mathematically, it is represented as: ``` a / sin A = b / sin B = c / sin C ``` Where: a, b, and c are the lengths of the sides of the triangle. A, B, and C are the angles opposite those sides, respectively.

When to use the Sine Rule: When you are given two angles and one side (AAS or ASA). When you are given two sides and an angle opposite one of them (SSA - ambiguous case, be mindful of two possible solutions).

Example 1: Finding an Unknown Side Suppose a farmer in KwaZulu-Natal wants to fence off a triangular piece of land. He knows two angles of the triangle are 60° and 80°, and the side opposite the 60° angle is 50 meters long. How long is the side opposite the 80° angle?

Solution: Let A = 60°, B = 80°, a = 50 m. We want to find b.

Using the Sine Rule: a / sin A = b / sin B Substituting the values: 50 / sin 60° = b / sin 80° Solving for b: b = (50 * sin 80°) / sin 60° Calculating: b ≈ (50 * 0.9848) / 0.8660 ≈ 56.85 meters Therefore, the side opposite the 80° angle is approximately 56.85 meters long.

Example 2: Finding an Unknown Angle A surveyor in Gauteng needs to determine the angle of a triangular plot of land. Two sides of the triangle are 100 m and 80 m, and the angle opposite the 100 m side is 70°. What is the angle opposite the 80 m side?

Solution: Let a = 100 m, b = 80 m, A = 70°. We want to find

B. Using the Sine Rule: a / sin A = b / sin B Substituting the values: 100 / sin 70° = 80 / sin B Solving for sin B: sin B = (80 * sin 70°) / 100 Calculating: sin B ≈ (80 * 0.9397) / 100 ≈ 0.7518 Finding B: B = arcsin(0.7518) ≈ 48.75° Therefore, the angle opposite the 80 m side is approximately 48.75°. 2.

2. Cosine Rule The Cosine Rule relates the lengths of the sides of a triangle to the cosine of one of its angles.

It is expressed as: ``` a² = b² + c² - 2bc cos A b² = a² + c² - 2ac cos B c² = a² + b² - 2ab cos C ``` Where: a, b, and c are the lengths of the sides of the triangle. A, B, and C are the angles opposite those sides, respectively.

When to use the Cosine Rule: When you are given three sides (SSS) and need to find an angle. When you are given two sides and the included angle (SAS) and need to find the third side.

Example 3: Finding an Unknown Side Two roads in Limpopo intersect at an angle of 110°. A truck travels 150 km on one road and another truck travels 200 km on the other road. How far apart are the two trucks?

Solution: Let b = 150 km, c = 200 km, A = 110°. We want to find a.

Using the Cosine Rule: a² = b² + c² - 2bc cos A Substituting the values: a² = (150)² + (200)² - 2(150)(200) cos 110° Calculating: a² = 22500 + 40000 - 60000 * (-0.3420) a² = 62500 + 20520 = 83020 Finding a: a = √83020 ≈ 288.13 km Therefore, the two trucks are approximately 288.13 km apart.

Example 4: Finding an Unknown Angle A triangular garden in the Western Cape has sides of length 8 m, 10 m, and 12 m. Find the largest angle of the garden.

Solution: The largest angle is opposite the longest side. Let a = 12 m, b = 8 m, c = 10 m. We want to find

A. Using the Cosine Rule: a² = b² + c² - 2bc cos A Rearranging to solve for cos A: cos A = (b² + c² - a²) / (2bc)

Substituting the values: cos A = (8² + 10² - 12²) / (2 8 10)

Calculating: cos A = (64 + 100 - 144) / 160 = 20 / 160 = 0.125 Finding A: A = arccos(0.125) ≈ 82.82° Therefore, the largest angle of the garden is approximately 82.82°. 2.

3. Area Rule The Area Rule states that the area of a triangle can be calculated if you know the lengths of two sides and the included angle (the angle between those two sides).

The formula is: ``` Area = ½ab sin C = ½bc sin A = ½ac sin B ``` Where: a, b, and c are the lengths of the sides of the triangle. A, B, and C are the angles opposite those sides, respectively.

When to use the Area Rule: When you are given two sides and the included angle (SAS).

Example 5: Calculating Area A triangular plot of land in Mpumalanga has sides of length 40 m and 60 m, and the angle between these sides is 55°. Calculate the area of the plot.

Solution: Let a = 40 m, b = 60 m, C = 55°.

Using the Area Rule: Area = ½ab sin C Substituting the values: Area = ½ 40 60 * sin 55° Calculating: Area = 1200 * 0.8192 ≈ 983.04 m² Therefore, the area of the plot of land is approximately 983.04 m². Guided Practice (With Solutions)

Question 1: In triangle PQR, p = 12 cm, q = 15 cm, and angle R = 30°. Calculate the area of triangle PQ

R. Solution: We are given two sides (p and q) and the included angle (R).

Use the Area Rule: Area = ½pq sin R Substitute: Area = ½ 12 15 * sin 30° Calculate: Area = ½ 12 15 * 0.5 = 45 cm² Therefore, the area of triangle PQR is 45 cm².

Question 2: In triangle ABC, angle A = 40°, angle B = 60°, and side c = 20 m. Calculate the length of side a.

Solution: We are given two angles (A and B) and one side (c).

We can find angle C: C = 180° - A - B = 180° - 40° - 60° = 80°.