Triangle Trigonometry

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Answer Key & Explanations

Triangle Trigonometry · Grade 11 · Worksheet 3

1. A right triangle is positioned on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around this triangle. What are the coordinates of the center of this circumscribed circle? Answer: (3,4) Solution: We have a right triangle with vertices at A(0,0), B(6,0), and C(6,8). Since AB is along the x-axis from (0,0) to (6,0), and BC is vertical from (6,0) to (6,8), angle B is 90 degrees.
   Full step-by-step solution

   Step 1: Understand the problem We have a right triangle with vertices at A(0,0), B(6,0), and C(6,8). Since AB is along the x-axis from (0,0) to (6,0), and BC is vertical from (6,0) to (6,8), angle B is 90 degrees. The circle circumscribed around the triangle is the circumcircle, and its center is the circumcenter. Step 2: Recall property of circumcenter of a right triangle For a right triangle, the circumcenter is the midpoint of the hypotenuse. The hypotenuse is the side opposite the right angle. Right angle is at B(6,0), so hypotenuse is AC, connecting A(0,0) and C(6,8). Step 3: Find midpoint of AC Midpoint formula: x-coordinate = (0 + 6)/2 = 6/2 = 3 y-coordinate = (0 + 8)/2 = 8/2 = 4 Step 4: Conclusion The circumcenter is at (3,4). Step 5: Verification (optional check) The circumcenter should be equidistant from A, B, and C. Distance from (3,4) to A(0,0): sqrt((3-0)^2 + (4-0)^2) = sqrt(9+16) = sqrt(25) = 5 Distance from (3,4) to B(6,0): sqrt((3-6)^2 + (4-0)^2) = sqrt(9+16) = 5 Distance from (3,4) to C(6,8): sqrt((3-6)^2 + (4-8)^2) = sqrt(9+16) = 5 All distances equal to 5, so correct. Final answer: (3,4)

2. Charlotte is a surveyor mapping a triangular plot of land. She measures two sides of the triangle as 14 meters and 19 meters, and the angle between these two sides is 48°. She needs to determine the length of the third side to complete her map. What is the length of the third side, rounded to the nearest meter? Answer: 14 meters Solution: We are given two sides and the included angle, so we use the Law of Cosines. Let a = 14 m, b = 19 m, and the included angle C = 48°. The unknown side c is opposite angle C.
   Full step-by-step solution

   Step 1: We are given two sides and the included angle, so we use the Law of Cosines. Let a = 14 m, b = 19 m, and the included angle C = 48°. The unknown side c is opposite angle C. Step 2: Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C) Step 3: Substitute the values: c^2 = 14^2 + 19^2 - 2 * 14 * 19 * cos(48°) Step 4: Calculate each part: 14^2 = 196 19^2 = 361 2 * 14 * 19 = 532 cos(48°) ≈ 0.66913 Step 5: Compute: c^2 = 196 + 361 - 532 * 0.66913 c^2 = 557 - 355.98 c^2 ≈ 201.02 Step 6: Take the square root: c ≈ sqrt(201.02) ≈ 14.18 meters Step 7: Round to the nearest meter: 14 meters. The third side is approximately 14 meters.

3. In triangle ABC, side a = 16, side b = 11, and angle C = 60°. Find side c. Answer: 14 Solution: Use the Law of Cosines: c² = a² + b² - 2ab cos(C) Substitute the given values: c² = 16² + 11² - 2(16)(11) cos(60°) Calculate squares: c² = 256 + 121 - 2(16)(11)(0.5) Simplify: c² = 377 - 352(0.5) = 377 - 176 Subtract: c² = 201 Take square root: c = √201 ≈ 14.177 Round to nearest whole number: c…
   Full step-by-step solution

   Step 1: Use the Law of Cosines: c² = a² + b² - 2ab cos(C) Step 2: Substitute the given values: c² = 16² + 11² - 2(16)(11) cos(60°) Step 3: Calculate squares: c² = 256 + 121 - 2(16)(11)(0.5) Step 4: Simplify: c² = 377 - 352(0.5) = 377 - 176 Step 5: Subtract: c² = 201 Step 6: Take square root: c = √201 ≈ 14.177 Step 7: Round to nearest whole number: c = 14 The answer is 14.

4. 2sin(75°)cos(75°) = ? Answer: 0.5 Solution: Recognize the double-angle identity: sin(2θ) = 2sinθcosθ Apply the identity: 2sin(75°)cos(75°) = sin(2 × 75°) Calculate the angle: 2 × 75° = 150° Evaluate sin(150°) = 0.5 Therefore, 2sin(75°)cos(75°) = 0.5 The answer is 0.5.
   Full step-by-step solution

   Step 1: Recognize the double-angle identity: sin(2θ) = 2sinθcosθ Step 2: Apply the identity: 2sin(75°)cos(75°) = sin(2 × 75°) Step 3: Calculate the angle: 2 × 75° = 150° Step 4: Evaluate sin(150°) = 0.5 Step 5: Therefore, 2sin(75°)cos(75°) = 0.5 The answer is 0.5.

5. sin(π/4)cos(π/12) + cos(π/4)sin(π/12) = ? Answer: √3/2 Solution: Recognize the trigonometric identity: sin(A)cos(B) + cos(A)sin(B) = sin(A+B) Identify A = π/4 and B = π/12 Calculate A + B = π/4 + π/12 = 3π/12 + π/12 = 4π/12 = π/3 The expression simplifies to sin(π/3) 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/4 and B = π/12 Step 3: Calculate A + B = π/4 + π/12 = 3π/12 + π/12 = 4π/12 = π/3 Step 4: The expression simplifies to sin(π/3) Step 5: sin(π/3) = √3/2 The answer is √3/2.

6. sin(π/3) + cos(π/6) = ? Answer: √3 Solution: Recall the value of sin(π/3) We know π/3 radians is 60 degrees. The sine of 60 degrees is √3/2. So, sin(π/3) = √3/2.
   Full step-by-step solution

   Step 1: Recall the value of sin(π/3) We know π/3 radians is 60 degrees. The sine of 60 degrees is √3/2. So, sin(π/3) = √3/2. Step 2: Recall the value of cos(π/6) We know π/6 radians is 30 degrees. The cosine of 30 degrees is √3/2. So, cos(π/6) = √3/2. Step 3: Add the two values sin(π/3) + cos(π/6) = √3/2 + √3/2. Step 4: Combine the terms Both terms have the same denominator, so we add the numerators: (√3 + √3)/2 = (2√3)/2. Step 5: Simplify (2√3)/2 = √3. Final Answer: √3

7. sin(2x) = 2sin(x)cos(x) for x = 45° Answer: A. true Solution: Write down the left-hand side (LHS) of the equation. LHS = sin(2x) = sin(2 * 45°) = sin(90°) Recall the value of sin(90°). sin(90°) = 1 So LHS = 1.
   Full step-by-step solution

   Let's check if sin(2x) = 2sin(x)cos(x) for x = 45°. Step 1: Write down the left-hand side (LHS) of the equation. LHS = sin(2x) = sin(2 * 45°) = sin(90°) Step 2: Recall the value of sin(90°). sin(90°) = 1 So LHS = 1. Step 3: Write down the right-hand side (RHS) of the equation. RHS = 2sin(x)cos(x) = 2sin(45°)cos(45°) Step 4: Recall the values of sin(45°) and cos(45°). sin(45°) = √2 / 2 cos(45°) = √2 / 2 Step 5: Substitute these values into the RHS. RHS = 2 * (√2 / 2) * (√2 / 2) Step 6: Simplify the expression. First, 2 * (√2 / 2) = √2 So RHS = √2 * (√2 / 2) = (√2 * √2) / 2 Step 7: Simplify further. √2 * √2 = 2 So RHS = 2 / 2 = 1 Step 8: Compare LHS and RHS. LHS = 1 RHS = 1 Since LHS = RHS, the equation is true. Therefore, the statement is correct.

8. Charlotte is a civil engineer designing a triangular support structure for a bridge. She measures two sides of the triangular frame: one side is 28 meters long, another side is 36 meters long, and the angle between these two sides is 115 degrees. Charlotte needs to calculate the length of the third side to order the correct amount of steel beams. What is the length of the third side, rounded to the nearest meter? Answer: 54 meters Solution: Identify the given values. Let side a = 28 m, side b = 36 m, and the included angle C = 115 degrees. We need side c (opposite angle C).
   Full step-by-step solution

   Step 1: Identify the given values. Let side a = 28 m, side b = 36 m, and the included angle C = 115 degrees. We need side c (opposite angle C). Step 2: Apply the Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C). Step 3: Substitute the known values: c^2 = 28^2 + 36^2 - 2(28)(36) cos(115°). Step 4: Calculate each part: - 28^2 = 784 - 36^2 = 1296 - 784 + 1296 = 2080 - 2(28)(36) = 2 * 1008 = 2016 - cos(115°) = cos(180° - 65°) = -cos(65°) ≈ -0.4226 Step 5: Substitute the cosine value: c^2 = 2080 - 2016 * (-0.4226) = 2080 + 2016 * 0.4226. Step 6: Compute 2016 * 0.4226 = 852.4416. Step 7: Add: c^2 = 2080 + 852.4416 = 2932.4416. Step 8: Take the square root: c = sqrt(2932.4416) ≈ 54.15 meters. Step 9: Round to the nearest meter: c ≈ 54 meters. The answer is 54 meters.