Multiple Angle Trigonometry

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Multiple Angle Trigonometry · Grade 12 · Worksheet 2

1. An engineer is designing a roller coaster track where the vertical position of the track follows the function y(x) = 3sin(2x) + 4cos(2x), where x is the horizontal distance in meters from the starting point. To ensure proper banking at turns, she needs to find all points where the track reaches exactly 2 meters above the reference line. Solve the equation 3sin(2x) + 4cos(2x) = 2 for x in the interval [0, 2π]. Answer: x = π/4, 3π/4, 5π/4, 7π/4 Solution: To solve equations of the form A·sin(θ) + B·cos(θ) = C, we can use the identity that allows us to combine sine and cosine terms into a single trigonometric function.
   Full step-by-step solution

   To solve equations of the form A·sin(θ) + B·cos(θ) = C, we can use the identity that allows us to combine sine and cosine terms into a single trigonometric function. This involves finding an amplitude and phase shift that transforms the left side into R·sin(θ + α) or R·cos(θ + α), where R is calculated using the Pythagorean theorem. Once in this form, we can solve using inverse trigonometric functions and consider all solutions within the given interval.

2. A triangular garden is bounded by three straight paths. The paths form a triangle with vertices at coordinates A(0,0), B(8,0), and C(4,6). A circular fountain is to be placed at the incenter of this triangle. What are the coordinates of the fountain's center? Answer: (4,2) Solution: The incenter of a triangle is the point where all three angle bisectors intersect, and it's equidistant from all three sides.
   Full step-by-step solution

   The incenter of a triangle is the point where all three angle bisectors intersect, and it's equidistant from all three sides. This point can be found using the formula that involves the triangle's vertices and side lengths, or by finding the intersection of any two angle bisectors. In coordinate geometry, this involves calculating distances and using angle bisector properties.

3. An engineer is designing a suspension bridge where the cable shape follows the curve y = 3sin(2x) + 4cos(2x). To determine the maximum tension points, she needs to find all values of x between 0 and 2π where the cable reaches its highest point. Solve the trigonometric equation to find these critical points. Answer: x = π/8, 5π/8, 9π/8, 13π/8 Solution: In bridge design and other engineering applications, finding maximum and minimum points of trigonometric functions is crucial for structural analysis.
   Full step-by-step solution

   In bridge design and other engineering applications, finding maximum and minimum points of trigonometric functions is crucial for structural analysis. The derivative of a function gives us the slope at any point, and where this slope equals zero, we find critical points that could be maximums, minimums, or inflection points. For trigonometric functions with transformed arguments, we can use identities to rewrite the equation in a more solvable form, then apply inverse trigonometric functions while considering the periodic nature of the solutions.

4. 2tan(4x) - 2√3 = 0 for x ∈ [0, 2π] Answer: π/12, π/3, 7π/12, 5π/6, 13π/12, 4π/3, 19π/12, 11π/6 Solution: Isolate tan(4x). 2tan(4x) - 2√3 = 0 2tan(4x) = 2√3 tan(4x) = √3 Find the reference angle. tan(θ) = √3 when θ = π/3 (60°) in the first quadrant.
   Full step-by-step solution

   Step 1: Isolate tan(4x). 2tan(4x) - 2√3 = 0 2tan(4x) = 2√3 tan(4x) = √3 Step 2: Find the reference angle. tan(θ) = √3 when θ = π/3 (60°) in the first quadrant. Since tangent is positive in quadrants I and III, the general solutions for 4x are: 4x = π/3 + πk, where k is an integer. Step 3: Solve for x. x = π/12 + πk/4 Step 4: Find all solutions in [0, 2π]. For k = 0: x = π/12 For k = 1: x = π/12 + π/4 = π/12 + 3π/12 = 4π/12 = π/3 For k = 2: x = π/12 + π/2 = π/12 + 6π/12 = 7π/12 For k = 3: x = π/12 + 3π/4 = π/12 + 9π/12 = 10π/12 = 5π/6 For k = 4: x = π/12 + π = π/12 + 12π/12 = 13π/12 For k = 5: x = π/12 + 5π/4 = π/12 + 15π/12 = 16π/12 = 4π/3 For k = 6: x = π/12 + 3π/2 = π/12 + 18π/12 = 19π/12 For k = 7: x = π/12 + 7π/4 = π/12 + 21π/12 = 22π/12 = 11π/6 For k = 8: x = π/12 + 2π = π/12 + 24π/12 = 25π/12, which is greater than 2π. Step 5: List all solutions in increasing order. π/12, π/3, 7π/12, 5π/6, 13π/12, 4π/3, 19π/12, 11π/6 Final answer: π/12, π/3, 7π/12, 5π/6, 13π/12, 4π/3, 19π/12, 11π/6

5. Find the smallest positive solution in radians to the equation: 4sin(3x) - 2√3 = 0 Answer: 0.349 Solution: 4 sin(3x) - 2√3 = 0 Isolate sin(3x) Add 2√3 to both sides: 4 sin(3x) = 2√3 Divide both sides by 4: sin(3x) = (2√3) / 4 sin(3x) = √3 / 2 General solution for 3x We know sin(θ) = √3 / 2 when θ = π/3 + 2πn or θ = 2π/3 + 2πn, where n is an integer.
   Full step-by-step solution

   Let's solve step by step. We have: 4 sin(3x) - 2√3 = 0 **Step 1: Isolate sin(3x)** Add 2√3 to both sides: 4 sin(3x) = 2√3 Divide both sides by 4: sin(3x) = (2√3) / 4 Simplify: sin(3x) = √3 / 2 **Step 2: General solution for 3x** We know sin(θ) = √3 / 2 when θ = π/3 + 2πn or θ = 2π/3 + 2πn, where n is an integer. So: 3x = π/3 + 2πn or 3x = 2π/3 + 2πn **Step 3: Solve for x** Divide each equation by 3: Case 1: x = (π/3)/3 + (2πn)/3 x = π/9 + 2πn/3 Case 2: x = (2π/3)/3 + (2πn)/3 x = 2π/9 + 2πn/3 **Step 4: Find smallest positive x** Let n = 0: Case 1: x = π/9 ≈ 3.1416/9 ≈ 0.349 Case 2: x = 2π/9 ≈ 6.2832/9 ≈ 0.698 The smallest positive is π/9 ≈ 0.349. **Final answer:** x = π/9 ≈ 0.349

6. 4tan(3x) - 4√3 = 0 for x ∈ [0, 2π) Answer: π/9, 4π/9, 7π/9, 10π/9, 13π/9, 16π/9 Solution: Isolate tan(3x): 4tan(3x) - 4√3 = 0 → 4tan(3x) = 4√3 → tan(3x) = √3. Find the reference angle: tan(θ) = √3 when θ = π/3 (60°) in the first quadrant.
   Full step-by-step solution

   Step 1: Isolate tan(3x): 4tan(3x) - 4√3 = 0 → 4tan(3x) = 4√3 → tan(3x) = √3. Step 2: Find the reference angle: tan(θ) = √3 when θ = π/3 (60°) in the first quadrant. Since tangent is positive in quadrants I and III, the general solutions for 3x are: 3x = π/3 + πk, where k is an integer. Step 3: Solve for x: x = π/9 + πk/3. Step 4: Find all solutions in [0, 2π): For k = 0: x = π/9 For k = 1: x = π/9 + π/3 = π/9 + 3π/9 = 4π/9 For k = 2: x = π/9 + 2π/3 = π/9 + 6π/9 = 7π/9 For k = 3: x = π/9 + π = π/9 + 9π/9 = 10π/9 For k = 4: x = π/9 + 4π/3 = π/9 + 12π/9 = 13π/9 For k = 5: x = π/9 + 5π/3 = π/9 + 15π/9 = 16π/9 For k = 6: x = π/9 + 2π = π/9 + 18π/9 = 19π/9, which is greater than 2π (18π/9), so stop. Step 5: List all solutions in increasing order: π/9, 4π/9, 7π/9, 10π/9, 13π/9, 16π/9. Final answer: π/9, 4π/9, 7π/9, 10π/9, 13π/9, 16π/9.

7. Mere is analyzing the path of a laser beam reflecting off a series of mirrors. The beam's angle of deviation from a straight line is given by the equation 2 sin(3θ) = √3, where θ is the angle of the beam measured in degrees. Find all possible values of θ in the interval [0°, 180°] that satisfy this equation. Answer: θ = 20°, 40°, 140°, 160° Solution: Isolate sin(3θ). 2 sin(3θ) = √3 implies sin(3θ) = √3/2. The reference angle for sin(3θ) = √3/2 is 60°, since sin(60°) = √3/2.
   Full step-by-step solution

   Step 1: Isolate sin(3θ). 2 sin(3θ) = √3 implies sin(3θ) = √3/2. Step 2: The reference angle for sin(3θ) = √3/2 is 60°, since sin(60°) = √3/2. Step 3: Since sine is positive in quadrants I and II, the general solutions for 3θ are: 3θ = 60° + 360°k or 3θ = 120° + 360°k, where k is an integer. Step 4: Solve for θ: θ = 20° + 120°k or θ = 40° + 120°k. Step 5: Find values in [0°, 180°] by testing integer k: For k = 0: θ = 20° and 40° (both in range). For k = 1: θ = 20° + 120° = 140° and 40° + 120° = 160° (both in range). For k = -1: θ = 20° - 120° = -100° and 40° - 120° = -80° (not in range). For k = 2: θ = 20° + 240° = 260° and 40° + 240° = 280° (not in range). Step 6: The solutions in [0°, 180°] are θ = 20°, 40°, 140°, 160°. The answer is θ = 20°, 40°, 140°, 160°.

8. 3tan(5x) - 3√3 = 0 for x ∈ [0, π]. Answer: π/15, 4π/15, 7π/15, 2π/3, 13π/15 Solution: Isolate tan(5x): 3tan(5x) - 3√3 = 0 → 3tan(5x) = 3√3 → tan(5x) = √3. Find the reference angle: tan(θ) = √3 when θ = π/3 (since tan(π/3) = √3). General solution for tan(θ) = √3: θ = π/3 + πk, where k is any integer.
   Full step-by-step solution

   Step 1: Isolate tan(5x): 3tan(5x) - 3√3 = 0 → 3tan(5x) = 3√3 → tan(5x) = √3. Step 2: Find the reference angle: tan(θ) = √3 when θ = π/3 (since tan(π/3) = √3). Step 3: General solution for tan(θ) = √3: θ = π/3 + πk, where k is any integer. Step 4: Substitute 5x for θ: 5x = π/3 + πk. Step 5: Solve for x: x = π/15 + πk/5. Step 6: Find all solutions in [0, π]: For k = 0: x = π/15. For k = 1: x = π/15 + π/5 = π/15 + 3π/15 = 4π/15. For k = 2: x = π/15 + 2π/5 = π/15 + 6π/15 = 7π/15. For k = 3: x = π/15 + 3π/5 = π/15 + 9π/15 = 10π/15 = 2π/3. For k = 4: x = π/15 + 4π/5 = π/15 + 12π/15 = 13π/15. For k = 5: x = π/15 + 5π/5 = π/15 + π = 16π/15, which is greater than π, so stop. Step 7: List all solutions in [0, π]: π/15, 4π/15, 7π/15, 2π/3, 13π/15. Final answer: π/15, 4π/15, 7π/15, 2π/3, 13π/15.