Trigonometric Identities

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Answer Key & Explanations

Trigonometric Identities · Grade 12 · Worksheet 3

1. A civil engineer is designing a suspension bridge where the main cable forms a parabolic curve described by the function f(x) = ax² + bx + c. The cable passes through points (-100, 50), (0, 10), and (100, 50) relative to the bridge's center. The engineer needs to verify that the cable's shape satisfies the trigonometric identity sin²θ + cos²θ = 1 at the point where the cable makes a 30° angle with the horizontal. Find the value of sin²θ + cos²θ at this point to confirm the identity holds. Answer: 1 Solution: (-100, 50), (0, 10), (100, 50) Equation: f(x) = a x^2 + b x + c From (0, 10): f(0) = a*0 + b*0 + c = 10 So c = 10.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Find the parabolic equation** We have points: (-100, 50), (0, 10), (100, 50) Equation: f(x) = a x^2 + b x + c From (0, 10): f(0) = a*0 + b*0 + c = 10 So c = 10. From (-100, 50): a*(10000) + b*(-100) + 10 = 50 10000a - 100b = 40 … (1) From (100, 50): a*(10000) + b*(100) + 10 = 50 10000a + 100b = 40 … (2) Add (1) and (2): 20000a = 80 a = 80 / 20000 = 8 / 2000 = 4 / 1000 = 0.004 From (2): 10000*(0.004) + 100b = 40 40 + 100b = 40 100b = 0 b = 0 So f(x) = 0.004 x^2 + 10 --- **Step 2: Find slope at a point where the cable makes 30° with horizontal** Slope = f'(x) = 0.008 x Let slope = tan θ = tan 30° = 1 / sqrt(3) ≈ 0.57735 So 0.008 x = 1 / sqrt(3) x = 1 / (0.008 * sqrt(3)) = 125 / sqrt(3) --- **Step 3: Find sin²θ + cos²θ** We know the trigonometric identity: sin²θ + cos²θ = 1 for any θ. So even without knowing the exact point on the cable, the identity holds. --- **Step 4: Conclusion** The problem says: "verify that the cable's shape satisfies the trigonometric identity sin²θ + cos²θ = 1 at the point where the cable makes a 30° angle with the horizontal." Since sin²θ + cos²θ = 1 is always true for any angle θ, the answer is 1. --- **Final answer:** 1

2. An engineer is designing a roller coaster with a drop that follows the path y = 25sin(x/10) + 15cos(x/10), where y is the height in meters and x is the horizontal distance in meters from the start of the drop. To ensure rider safety, she needs to verify that this path can be rewritten in the form y = Rsin(x/10 + α), where R > 0 and α is a phase shift. Find the amplitude R of this transformed function, which represents the maximum deviation from the average height. Answer: 29.154759474 Solution: The original function is y = 25sin(x/10) + 15cos(x/10) To write this in the form y = Rsin(x/10 + α), we use the identity Rsin(θ + α) = Rsinθcosα + Rcosθsinα Comparing coefficients: Rcosα = 25 and Rsinα = 15 To find R, square both equations and add them: (Rcosα)² + (Rsinα)² = 25² + 15² R²(cos²α +…
   Full step-by-step solution

   Step 1: The original function is y = 25sin(x/10) + 15cos(x/10) Step 2: To write this in the form y = Rsin(x/10 + α), we use the identity Rsin(θ + α) = Rsinθcosα + Rcosθsinα Step 3: Comparing coefficients: Rcosα = 25 and Rsinα = 15 Step 4: To find R, square both equations and add them: (Rcosα)² + (Rsinα)² = 25² + 15² Step 5: R²(cos²α + sin²α) = 625 + 225 Step 6: Since cos²α + sin²α = 1, we have R² = 850 Step 7: R = sqrt(850) = sqrt(25 × 34) = 5sqrt(34) Step 8: 5sqrt(34) ≈ 5 × 5.8309518948 ≈ 29.154759474 The amplitude is approximately 29.15 meters.

3. Verify: (sec²θ - 1) / (tan²θ + 1) = sin²θ. Answer: sin²θ Solution: Start with the left side: (sec²θ - 1) / (tan²θ + 1). Use identities: sec²θ = 1/cos²θ, tan²θ = sin²θ/cos²θ. Substitute: (1/cos²θ - 1) / (sin²θ/cos²θ + 1).
   Full step-by-step solution

   Start with the left side: (sec²θ - 1) / (tan²θ + 1). Step 1: Use identities: sec²θ = 1/cos²θ, tan²θ = sin²θ/cos²θ. Step 2: Substitute: (1/cos²θ - 1) / (sin²θ/cos²θ + 1). Step 3: Combine numerator: (1 - cos²θ)/cos²θ = sin²θ/cos²θ (since 1 - cos²θ = sin²θ). Step 4: Combine denominator: (sin²θ + cos²θ)/cos²θ = 1/cos²θ (since sin²θ + cos²θ = 1). Step 5: The left side becomes (sin²θ/cos²θ) ÷ (1/cos²θ) = (sin²θ/cos²θ) * (cos²θ/1) = sin²θ. Thus, left side equals right side, verifying the identity.

4. An electrical engineer is analyzing alternating current in a circuit where the voltage is given by V(t) = 3sin(2t) + 4cos(2t) volts. To design proper safety components, she needs to verify that this voltage function can be rewritten in the form V(t) = Rsin(2t + α), where R > 0 represents the amplitude and α is the phase shift. Using trigonometric identities, determine the values of R and α that satisfy this transformation. Answer: R = 5, α = arctan(4/3) Solution: When working with alternating current circuits, engineers often need to express combinations of sine and cosine functions as a single sine function with amplitude and phase shift.
   Full step-by-step solution

   When working with alternating current circuits, engineers often need to express combinations of sine and cosine functions as a single sine function with amplitude and phase shift. This transformation uses trigonometric identities to find the amplitude using the square root of the sum of squares of coefficients, and the phase shift using the arctangent of the ratio of coefficients. This representation makes it easier to analyze the maximum voltage and timing relationships in electrical systems.

5. Verify: (sin⁴θ - cos⁴θ) / (sin²θ - cos²θ) = 1. Answer: 1 Solution: Factor the numerator sin⁴θ - cos⁴θ as (sin²θ - cos²θ)(sin²θ + cos²θ). The expression becomes [(sin²θ - cos²θ)(sin²θ + cos²θ)] / (sin²θ - cos²θ). Use the Pythagorean identity sin²θ + cos²θ = 1.
   Full step-by-step solution

   Step 1: Factor the numerator sin⁴θ - cos⁴θ as (sin²θ - cos²θ)(sin²θ + cos²θ). Step 2: The expression becomes [(sin²θ - cos²θ)(sin²θ + cos²θ)] / (sin²θ - cos²θ). Step 3: Cancel the common factor (sin²θ - cos²θ), provided sin²θ ≠ cos²θ. Step 4: Use the Pythagorean identity sin²θ + cos²θ = 1. Step 5: The result is 1. The answer is 1.

6. Kaia is analyzing the following trigonometric identity involving a right triangle drawn on a coordinate plane. The triangle has its hypotenuse of length 13 units, with one leg along the positive x-axis from the origin to the point (12, 0), and the other leg vertical from (12, 0) to (12, 5). Let angle theta be the angle at the origin between the hypotenuse and the x-axis. Verify the identity: (1 - sin^2(theta)) / (cos(theta)) = cos(theta). Answer: cos(theta) = 12/13, identity holds Solution: From the triangle, the hypotenuse is 13, the adjacent side (along x-axis) is 12, and the opposite side (vertical) is 5. The left side of the identity is (1 - sin^2(theta)) / cos(theta).
   Full step-by-step solution

   Step 1: From the triangle, the hypotenuse is 13, the adjacent side (along x-axis) is 12, and the opposite side (vertical) is 5. Thus, sin(theta) = opposite/hypotenuse = 5/13, and cos(theta) = adjacent/hypotenuse = 12/13. Step 2: The left side of the identity is (1 - sin^2(theta)) / cos(theta). Substitute sin(theta) = 5/13: sin^2(theta) = (5/13)^2 = 25/169. Then 1 - sin^2(theta) = 1 - 25/169 = (169/169 - 25/169) = 144/169. Step 3: Now divide by cos(theta) = 12/13: (144/169) / (12/13) = (144/169) * (13/12) = (144 * 13) / (169 * 12). Step 4: Simplify: 144/12 = 12, and 13/169 = 1/13, so the expression becomes 12 * (1/13) = 12/13. Step 5: The right side of the identity is cos(theta) = 12/13. Since both sides equal 12/13, the identity is verified. The answer is cos(theta) = 12/13, identity holds.