Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Apply the Sine Rule to solve triangles.
• Apply the Cosine Rule to solve triangles.
• Calculate the area of a triangle using the formula Area = ½ab sin(C).
• Solve practical problems using these rules.
• Decide which rule is appropriate for a given problem.

Lesson summary

This week, we delve deeper into Trigonometry, building upon the fundamental concepts of sine, cosine, and tangent that you learned previously. Our focus will be on understanding and applying the Sine Rule, the Cosine Rule, and calculating the Area of a Triangle using trigonometric ratios even when the height is not directly provided. This knowledge is crucial for solving various real-world problems, from surveying land to navigation. Think about town planners designing residential layouts, farmers calculating field sizes, or even civil engineers determining the angles and lengths needed for bridge construction. Trigonometry is a foundational skill that opens doors to numerous career paths.

Lesson notes

2.1 The Sine Rule The Sine Rule establishes a relationship between the sides of a triangle and the sines of their opposite angles. For any triangle ABC, with sides a, b, and c opposite to angles A, B, and C respectively: ``` a / sin(A) = b / sin(B) = c / sin(C) ``` This rule is particularly useful when you know: Two angles and one side (AAS) Two sides and a non-included angle (SSA) – be mindful of the ambiguous case with SSA!* Important Notes on the Ambiguous Case (SSA): When given two sides and a non-included angle, there might be two possible solutions, one solution, or no solution. This ambiguity arises because the given information can sometimes form two different triangles. Always check for a second possible solution by considering the supplementary angle (180° - calculated angle).

Example 1 (AAS): In triangle PQR, angle P = 40°, angle Q = 65°, and side r = 12 cm. Find the length of side p.

Solution: We have two angles and a side (AAS). We want to find side 'p', so we use the Sine Rule involving 'p' and 'r': p / sin(P) = r / sin(R) First, find angle R: R = 180° - P - Q = 180° - 40° - 65° = 75° Substitute the known values: p / sin(40°) = 12 / sin(75°)

Solve for p: p = (12 * sin(40°)) / sin(75°) ≈ 7.97 cm Example 2 (SSA): In triangle ABC, side a = 8 cm, side b = 10 cm, and angle A = 30°. Find angle

B. Solution: We have two sides and a non-included angle (SSA).

Use the Sine Rule: a / sin(A) = b / sin(B)

Substitute the known values: 8 / sin(30°) = 10 / sin(B)

Solve for sin(B): sin(B) = (10 * sin(30°)) / 8 = 0.625 Find angle B: B = arcsin(0.625) ≈ 38.68° Check for a second solution: B' = 180° - 38.68° = 141.32°. Since A + B' = 30° + 141.32° = 171.32° < 180°, a second solution is possible.

Therefore, B ≈ 38.68° or B ≈ 141.32°. You'll need more information or context to determine which solution is correct (or if both are valid). 2.2 The Cosine Rule The Cosine Rule relates the sides of a triangle to the cosine of one of its angles. For any triangle ABC, with sides a, b, and c opposite to angles A, B, and C respectively: ``` a² = b² + c² - 2bc * cos(A) b² = a² + c² - 2ac * cos(B) c² = a² + b² - 2ab * cos(C) ``` The Cosine Rule is particularly useful when you know: Two sides and the included angle (SAS)

All three sides (SSS)

Example 3 (SAS): In triangle XYZ, side x = 5 m, side y = 7 m, and angle Z = 60°. Find the length of side z.

Solution: We have two sides and the included angle (SAS). We want to find side 'z', so we use the Cosine Rule: z² = x² + y² - 2xy * cos(Z)

Substitute the known values: z² = 5² + 7² - 2 5 7 * cos(60°)

Calculate: z² = 25 + 49 - 70 * (1/2) = 39 Solve for z: z = √39 ≈ 6.25 m Example 4 (SSS): In triangle KLM, side k = 6 cm, side l = 8 cm, and side m = 12 cm. Find the measure of angle

K. Solution: We have all three sides (SSS). We want to find angle K, so we use the Cosine Rule (rearranged): cos(K) = (l² + m² - k²) / (2lm)

Substitute the known values: cos(K) = (8² + 12² - 6²) / (2 8 12)

Calculate: cos(K) = (64 + 144 - 36) / 192 = 172 / 192 = 0.8958 Solve for K: K = arccos(0.8958) ≈ 26.4° 2.3 Area of a Triangle Traditionally, the area of a triangle is calculated as (1/2) base height.

However, we can also calculate the area using trigonometry when we know two sides and the included angle: ``` Area = (1/2) ab sin(C) Area = (1/2) ac sin(B) Area = (1/2) bc sin(A) ``` Example 5: In triangle DEF, side d = 10 cm, side e = 15 cm, and angle F = 45°. Find the area of the triangle.

Solution: We have two sides and the included angle.

Use the area formula: Area = (1/2) de sin(F)

Substitute the known values: Area = (1/2) 10 15 * sin(45°)

Calculate: Area = 75 * (√2 / 2) ≈ 53.03 cm² Guided Practice (With Solutions)

Question 1: In triangle ABC, angle A = 50°, angle B = 70°, and side c = 8 cm. Find the length of side a.

Solution: Identify the given information: AAS (Two angles and a side).

Use the Sine Rule: a / sin(A) = c / sin(C)

Find angle C: C = 180° - A - B = 180° - 50° - 70° = 60° Substitute: a / sin(50°) = 8 / sin(60°)

Solve for a: a = (8 * sin(50°)) / sin(60°) ≈ 7.07 cm

Commentary: We used the Sine Rule because we had two angles and a side. We first calculated the missing angle C before applying the rule to find the unknown side 'a'.

Question 2: In triangle PQR, side p = 7 cm, side q = 10 cm, and angle R = 55°. Find the length of side r.

Solution: Identify the given information: SAS (Two sides and the included angle).

Use the Cosine Rule: r² = p² + q² - 2pq * cos(R)

Substitute: r² = 7² + 10² - 2 7 10 * cos(55°)

Calculate: r² = 49 + 100 - 140 * cos(55°) ≈ 68.99 Solve for r: r = √68.99 ≈ 8.31 cm

Commentary: We used the Cosine Rule because we were given two sides and the included angle. The Cosine Rule directly allowed us to calculate the unknown side 'r'.

Question 3: In triangle XYZ, side x = 5 cm, side y = 8 cm, and angle Z = 40°. Find the area of the triangle.

Solution: Identify the given information: Two sides and the included angle.