Trigonometric Identities

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Answer Key & Explanations

Trigonometric Identities · Grade 12 · Worksheet 3

1. Liam is designing a suspension bridge where the main cable follows the curve y = 15sin(x/10) + 20, where y is the height in meters above the water and x is the horizontal distance in meters from the left tower. Engineers need to know the steepest slope of the cable to design proper connections. What is the maximum slope of the cable? Answer: 1.5 Solution: y = 15 sin(x/10) + 20 We want the maximum slope of the cable. Slope is the derivative dy/dx.
   Full step-by-step solution

   Step 1: Understand the problem The cable's curve is given by y = 15 sin(x/10) + 20 We want the maximum slope of the cable. Slope is the derivative dy/dx. Step 2: Find the derivative y = 15 sin(x/10) + 20 dy/dx = 15 * cos(x/10) * (1/10) dy/dx = (15/10) cos(x/10) dy/dx = 1.5 cos(x/10) Step 3: Interpret the slope expression Slope m(x) = 1.5 cos(x/10) The maximum possible slope occurs when cos(x/10) is maximum. The maximum value of cos(anything) is 1. Step 4: Find the maximum slope Maximum slope = 1.5 * 1 = 1.5 Step 5: Conclusion The steepest slope of the cable is 1.5 meters rise per meter horizontally. This occurs where cos(x/10) = 1, i.e., where x/10 = 2nπ for integer n. Final answer: 1.5

2. An architect is designing a curved roof for a modern art museum. The roof's cross-section follows the path y = 4sin(x) + 3cos(x), where y is the height in meters and x is the horizontal distance from the center. To determine the maximum stress points, the architect needs to rewrite this function in the form y = Rsin(x + α), where R > 0 and α is a phase shift. What are the values of R and α that satisfy this trigonometric identity? Answer: R = 5, α = arctan(3/4) Solution: Start with the given function: y = 4sin(x) + 3cos(x) To write in the form y = Rsin(x + α), use the identity: Rsin(x + α) = Rsin(x)cos(α) + Rcos(x)sin(α) Compare coefficients: Rcos(α) = 4 and Rsin(α) = 3 Find R using R² = (Rcos(α))² + (Rsin(α))² = 4² + 3² = 16 + 9 = 25 Therefore R = sqrt(25) = 5…
   Full step-by-step solution

   Step 1: Start with the given function: y = 4sin(x) + 3cos(x) Step 2: To write in the form y = Rsin(x + α), use the identity: Rsin(x + α) = Rsin(x)cos(α) + Rcos(x)sin(α) Step 3: Compare coefficients: Rcos(α) = 4 and Rsin(α) = 3 Step 4: Find R using R² = (Rcos(α))² + (Rsin(α))² = 4² + 3² = 16 + 9 = 25 Step 5: Therefore R = sqrt(25) = 5 Step 6: Find α using tan(α) = (Rsin(α))/(Rcos(α)) = 3/4 Step 7: Therefore α = arctan(3/4) The values are R = 5 and α = arctan(3/4).

3. Liam is designing a suspension bridge where the main cable forms a parabolic curve. The cable is anchored at two towers 200 meters apart, with the lowest point 50 meters below the tower tops. To verify the structural integrity, Liam needs to prove that the cable's shape follows the equation y = (1/250)x² - 50, where x is the horizontal distance from the center and y is the height relative to the lowest point. Using trigonometric identities, show that the angle θ between the cable and horizontal at a point 100 meters from the center satisfies tan²θ = (4x²)/(62500 + x⁴) for this parabolic shape. Answer: The derivative dy/dx = (2/250)x = x/125. At x = 100, dy/dx = 100/125 = 4/5. Then tanθ = dy/dx = 4/5, so tan²θ = 16/25. The given expression (4x²)/(62500 + x⁴) at x = 100 gives (4×10000)/(62500 + 100000000) = 40000/100062500 = 16/25. Both equal 16/25, verifying the identity. Solution: The slope of a curve at any point, given by its derivative, determines the angle the curve makes with the horizontal. This angle affects how forces are distributed throughout the structure.
   Full step-by-step solution

   In bridge design and other structural engineering applications, verifying mathematical relationships is crucial for safety. The slope of a curve at any point, given by its derivative, determines the angle the curve makes with the horizontal. This angle affects how forces are distributed throughout the structure. Trigonometric identities help engineers confirm that their mathematical models accurately represent physical reality, ensuring structures can withstand expected loads and environmental conditions.

4. An engineer is designing a roller coaster with a track section that follows the path y = 3sin(x) - 4cos(x), where y is the height in meters and x is the horizontal distance in meters. To ensure safety, she needs to verify that this path can be rewritten in the form y = Rsin(x - α) where R > 0. Find the exact values of R and α that satisfy this identity. Answer: R = 5, α = arctan(4/3) Solution: Step 1: We need to rewrite y = 3sin(x) - 4cos(x) in the form y = Rsin(x - α) Step 2: Using the sine subtraction formula: Rsin(x - α) = R[sin(x)cos(α) - cos(x)sin(α)] Step 3: Compare coefficients with y = 3sin(x) - 4cos(x): Rcos(α) = 3 Rsin(α) = 4 Step 4: To find R, square both equations and add…
   Full step-by-step solution

   Step 1: We need to rewrite y = 3sin(x) - 4cos(x) in the form y = Rsin(x - α) Step 2: Using the sine subtraction formula: Rsin(x - α) = R[sin(x)cos(α) - cos(x)sin(α)] Step 3: Compare coefficients with y = 3sin(x) - 4cos(x): Rcos(α) = 3 Rsin(α) = 4 Step 4: To find R, square both equations and add them: (Rcos(α))² + (Rsin(α))² = 3² + 4² R²(cos²(α) + sin²(α)) = 9 + 16 R²(1) = 25 R = 5 Step 5: To find α, divide the second equation by the first: (Rsin(α))/(Rcos(α)) = 4/3 tan(α) = 4/3 α = arctan(4/3) Step 6: Verify the signs: Rcos(α) = 5cos(α) = 3 (positive) and Rsin(α) = 5sin(α) = 4 (positive), so α is in the first quadrant. The answer is R = 5 and α = arctan(4/3).

5. 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.

6. Verify: (sin⁶θ + cos⁶θ) / (sin²θ + cos²θ) = 1 - 3 sin²θ cos²θ. Answer: Verified identity Solution: Start with the left side: (sin⁶θ + cos⁶θ) / (sin²θ + cos²θ). Since sin²θ + cos²θ = 1, the denominator is 1, so the expression simplifies to sin⁶θ + cos⁶θ. Now factor sin⁶θ + cos⁶θ as (sin²θ)³ + (cos²θ)³.
   Full step-by-step solution

   Start with the left side: (sin⁶θ + cos⁶θ) / (sin²θ + cos²θ). Since sin²θ + cos²θ = 1, the denominator is 1, so the expression simplifies to sin⁶θ + cos⁶θ. Now factor sin⁶θ + cos⁶θ as (sin²θ)³ + (cos²θ)³. Using sum of cubes: a³ + b³ = (a + b)(a² - ab + b²), with a = sin²θ, b = cos²θ. So sin⁶θ + cos⁶θ = (sin²θ + cos²θ)(sin⁴θ - sin²θ cos²θ + cos⁴θ). Since sin²θ + cos²θ = 1, this becomes sin⁴θ - sin²θ cos²θ + cos⁴θ. Rewrite sin⁴θ + cos⁴θ as (sin²θ + cos²θ)² - 2 sin²θ cos²θ = 1 - 2 sin²θ cos²θ. Thus the expression is (1 - 2 sin²θ cos²θ) - sin²θ cos²θ = 1 - 3 sin²θ cos²θ. This matches the right side. The identity is verified.

7. A right circular cone is inscribed in a sphere of radius 10 cm. The height of the cone is equal to the diameter of its base. Find the volume of the cone. Answer: 320π/3 cm³ Solution: When a cone is inscribed in a sphere, the vertices of the cone lie on the sphere's surface.
   Full step-by-step solution

   When a cone is inscribed in a sphere, the vertices of the cone lie on the sphere's surface. The relationship between the cone's dimensions and the sphere's radius can be found using coordinate geometry, placing the sphere's center at the origin and applying the distance formula to points on the cone that must satisfy the sphere's equation. This creates a system of equations that can be solved for the cone's dimensions.