Triangle Trigonometry

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Triangle Trigonometry · Grade 11 · Worksheet 1

1. Isabella is a marine archaeologist mapping a triangular underwater site. She measures two sides of the site as 18 meters and 25 meters, and the angle opposite the 18-meter side is 32°. She needs to determine whether this information defines one unique triangle, two possible triangles, or no triangle at all to plan her excavation grid. How many distinct triangles are possible with these measurements? Answer: Two Solution: Identify given values: side a = 18 m (opposite angle A = 32°), side b = 25 m (opposite angle B, unknown). Use Law of Sines: a / sin(A) = b / sin(B). Substitute: 18 / sin(32°) = 25 / sin(B).
   Full step-by-step solution

   Step 1: Identify given values: side a = 18 m (opposite angle A = 32°), side b = 25 m (opposite angle B, unknown). Step 2: Use Law of Sines: a / sin(A) = b / sin(B). Step 3: Substitute: 18 / sin(32°) = 25 / sin(B). Step 4: Calculate sin(32°) ≈ 0.5299. Then 18 / 0.5299 ≈ 33.97. Step 5: Set up: 33.97 = 25 / sin(B) → sin(B) = 25 / 33.97 ≈ 0.7360. Step 6: Since sin(B) = 0.7360 < 1, there are two possible angles for B: B₁ ≈ arcsin(0.7360) ≈ 47.4°, and B₂ ≈ 180° - 47.4° = 132.6°. Step 7: Check each with angle A = 32°: - For B₁ = 47.4°, C₁ = 180° - 32° - 47.4° = 100.6° (valid, positive). - For B₂ = 132.6°, C₂ = 180° - 32° - 132.6° = 15.4° (valid, positive). Step 8: Both cases produce valid triangles, so two distinct triangles are possible. The answer is Two.

2. Emma is a marine biologist measuring the distance between two buoys in a triangular research zone. She knows that one side of the triangle between buoy A and buoy B is 40 meters, the side between buoy A and buoy C is 55 meters, and the angle at buoy A between these two sides is 45°. Emma needs to calculate the exact distance between buoy B and buoy C to plan her boat route. What is the distance between buoy B and buoy C? Answer: 5√(65 - 44√2) meters Solution: Identify the given information. Side a = 40 m (AB), side b = 55 m (AC), included angle C = 45° at buoy A. We need side c (BC).
   Full step-by-step solution

   Step 1: Identify the given information. Side a = 40 m (AB), side b = 55 m (AC), included angle C = 45° at buoy A. We need side c (BC). Step 2: Use the Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C) Step 3: Substitute the values: c^2 = 40^2 + 55^2 - 2 * 40 * 55 * cos(45°) Step 4: Calculate each term: 40^2 = 1600 55^2 = 3025 2 * 40 * 55 = 4400 cos(45°) = √2/2 Step 5: Substitute: c^2 = 1600 + 3025 - 4400 * (√2/2) c^2 = 4625 - 2200√2 Step 6: Factor out common factor 25: c^2 = 25 * (185 - 88√2) Step 7: Take the square root: c = √(25 * (185 - 88√2)) = 5√(185 - 88√2) Step 8: Simplify inside the radical: 185 = 25 * 7.4, but check if 185 - 88√2 can be written as a perfect square. Note that (a - b√2)^2 = a^2 + 2b^2 - 2ab√2. Try a = 11, b = 4: 11^2 + 2*4^2 = 121 + 32 = 153, not 185. Try a = 13, b = 2: 169 + 8 = 177, not 185. Try a = 5√? Not needed. Alternatively, note 185 = 25*7.4, but simpler: 185 - 88√2 = (√? )^2. Actually, let's verify if 185 - 88√2 = (11 - 4√2)^2? (11 - 4√2)^2 = 121 + 32 - 88√2 = 153 - 88√2, no. Try (13 - 2√2)^2 = 169 + 8 - 52√2 = 177 - 52√2, no. Try (5√? )... Since no perfect square, the answer is 5√(185 - 88√2) meters. But note the problem expects a simplified form. Since 185 = 5*37 and 88 = 8*11, no common factor simplifies further. However, note that 185 - 88√2 can be written as (√185)^2 - 2*(√185)*(4√2) + (4√2)^2? No. The simplest exact form is 5√(185 - 88√2). But to match typical textbook answer, we can rationalize differently: 185 - 88√2 = (√? )... Actually, note that 185 = 25*7.4, but better: 185 - 88√2 = (√185 - 4√2)^2? (√185 - 4√2)^2 = 185 + 32 - 8√370, no. So the answer remains 5√(185 - 88√2) meters. However, to simplify further: 185 = 5*37, 88 = 8*11, so no. The final answer is 5√(185 - 88√2) meters.

3. sin(π/4)cos(π/4) + cos(π/4)sin(π/4) = ? Answer: 1 Solution: Recognize that sin(A)cos(B) + cos(A)sin(B) = sin(A+B) Here A = π/4 and B = π/4 So the expression equals sin(π/4 + π/4) = sin(π/2) sin(π/2) = 1 Therefore, the answer is 1
   Full step-by-step solution

   Step 1: Recognize that sin(A)cos(B) + cos(A)sin(B) = sin(A+B) Step 2: Here A = π/4 and B = π/4 Step 3: So the expression equals sin(π/4 + π/4) = sin(π/2) Step 4: sin(π/2) = 1 Step 5: Therefore, the answer is 1

4. Ava is a civil engineer designing a triangular support structure for a pedestrian bridge. She knows that two sides of the triangular support measure 26 meters and 31 meters, and the angle opposite the 26-meter side is 41°. Ava needs to determine how many possible triangular configurations exist for the support structure to ensure it can be built safely. How many distinct triangles can be formed with these given measurements? Answer: Two distinct triangles Solution: Identify the given values: side a = 26 m (opposite angle A = 41°), side b = 31 m. We need to find possible values for angle B. Use the Law of Sines: a/sin(A) = b/sin(B) → 26/sin(41°) = 31/sin(B).
   Full step-by-step solution

   Step 1: Identify the given values: side a = 26 m (opposite angle A = 41°), side b = 31 m. We need to find possible values for angle B. Step 2: Use the Law of Sines: a/sin(A) = b/sin(B) → 26/sin(41°) = 31/sin(B). Step 3: Calculate sin(41°) ≈ 0.6561. So 26/0.6561 ≈ 39.63. Step 4: Set up the equation: 39.63 = 31/sin(B) → sin(B) = 31/39.63 ≈ 0.7822. Step 5: Since sin(B) = 0.7822 < 1, two angles satisfy this: B₁ = sin⁻¹(0.7822) ≈ 51.5° and B₂ = 180° - 51.5° = 128.5°. Step 6: Check if both are valid with A = 41°: - For B₁ = 51.5°, C₁ = 180° - 41° - 51.5° = 87.5° (valid, positive angle). - For B₂ = 128.5°, C₂ = 180° - 41° - 128.5° = 10.5° (valid, positive angle). Step 7: Both triangles satisfy the triangle angle sum property, so two distinct triangles are possible. The answer is two distinct triangles.

5. In triangle PQR, side p = 12, side q = 17, and angle R = 72°. Find side r. Answer: sqrt(433 - 408*cos(72°)) Solution: Use the Law of Cosines: r^2 = p^2 + q^2 - 2pq cos(R). Substitute p = 12, q = 17, R = 72°: r^2 = 12^2 + 17^2 - 2(12)(17) cos(72°). Calculate squares: 12^2 = 144, 17^2 = 289.
   Full step-by-step solution

   Step 1: Use the Law of Cosines: r^2 = p^2 + q^2 - 2pq cos(R). Step 2: Substitute p = 12, q = 17, R = 72°: r^2 = 12^2 + 17^2 - 2(12)(17) cos(72°). Step 3: Calculate squares: 12^2 = 144, 17^2 = 289. Step 4: Sum: 144 + 289 = 433. Step 5: Compute 2pq = 2(12)(17) = 408. Step 6: So r^2 = 433 - 408 cos(72°). Step 7: Take square root: r = sqrt(433 - 408 cos(72°)). The answer is sqrt(433 - 408 cos(72°)).

6. sin(π/12)cos(π/4) + cos(π/12)sin(π/4) = ? Answer: √3/2 Solution: Recognize the trigonometric identity: sin(A)cos(B) + cos(A)sin(B) = sin(A + B) Identify A = π/12 and B = π/4 Apply the identity: sin(π/12)cos(π/4) + cos(π/12)sin(π/4) = sin(π/12 + π/4) Add the angles: π/12 + π/4 = π/12 + 3π/12 = 4π/12 = π/3 Evaluate sin(π/3) = √3/2 The answer is √3/2.
   Full step-by-step solution

   Step 1: Recognize the trigonometric identity: sin(A)cos(B) + cos(A)sin(B) = sin(A + B) Step 2: Identify A = π/12 and B = π/4 Step 3: Apply the identity: sin(π/12)cos(π/4) + cos(π/12)sin(π/4) = sin(π/12 + π/4) Step 4: Add the angles: π/12 + π/4 = π/12 + 3π/12 = 4π/12 = π/3 Step 5: Evaluate sin(π/3) = √3/2 The answer is √3/2.

7. Noah is a surveyor mapping a triangular plot of land for a new park. He measures two sides of the plot: one side is 28 meters long, another side is 34 meters long, and the angle between these two sides is 115 degrees. Noah needs to find the area of the plot to determine how much sod is needed for landscaping. What is the area of the triangular plot in square meters? Answer: approximately 431.5 square meters Solution: Identify the given values: side a = 28 meters, side b = 34 meters, included angle C = 115 degrees. Use the formula for the area of a triangle with two sides and the included angle: Area = (1/2) * a * b * sin(C).
   Full step-by-step solution

   Step 1: Identify the given values: side a = 28 meters, side b = 34 meters, included angle C = 115 degrees. Step 2: Use the formula for the area of a triangle with two sides and the included angle: Area = (1/2) * a * b * sin(C). Step 3: Substitute the values: Area = (1/2) * 28 * 34 * sin(115 degrees). Step 4: Calculate the product of the sides: 28 * 34 = 952. Step 5: Multiply by 1/2: (1/2) * 952 = 476. Step 6: Find sin(115 degrees). Since 115 degrees is in quadrant II, sin(115) = sin(180 - 115) = sin(65 degrees). Using a calculator, sin(65 degrees) is approximately 0.9063. Step 7: Multiply: 476 * 0.9063 = 431.3988. Step 8: Round to one decimal place: approximately 431.4 square meters. Step 9: Therefore, the area of the triangular plot is approximately 431.4 square meters.