Trigonometric Identities

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Answer Key & Explanations

Trigonometric Identities · Grade 12 · Worksheet 2

1. A right circular cone has a height of 15 cm and a base radius of 8 cm. A plane parallel to the base cuts the cone, creating a smaller similar cone at the top with a height of 5 cm from the vertex. What is the volume of the frustum (the remaining portion between the two parallel planes)? Use π in your calculation. Answer: 2480π/3 Solution: Find the radius of the smaller cone using similarity. The ratio of heights is 5/15 = 1/3. Since the cones are similar, the radius ratio is also 1/3.
   Full step-by-step solution

   Step 1: Find the radius of the smaller cone using similarity. The ratio of heights is 5/15 = 1/3. Since the cones are similar, the radius ratio is also 1/3. Small cone radius = (1/3) × 8 = 8/3 cm. Step 2: Calculate the volume of the large cone. Volume = (1/3)πr²h = (1/3)π(8)²(15) = (1/3)π(64)(15) = (1/3)π(960) = 320π cm³. Step 3: Calculate the volume of the small cone. Volume = (1/3)πr²h = (1/3)π(8/3)²(5) = (1/3)π(64/9)(5) = (1/3)π(320/9) = 320π/27 cm³. Step 4: Subtract to find the volume of the frustum. Frustum volume = Large cone volume - Small cone volume = 320π - 320π/27 = (8640π/27 - 320π/27) = 8320π/27 = 2480π/3 cm³. The answer is 2480π/3.

2. 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).

3. Verify: (sin⁶θ - cos⁶θ) / (sin²θ - cos²θ) = 1 + sin²θ cos²θ. Answer: Verified identity Solution: Start with the left side: (sin⁶θ - cos⁶θ) / (sin²θ - cos²θ). Recognize the numerator as a difference of cubes: a³ - b³ = (a - b)(a² + ab + b²). Here a = sin²θ, b = cos²θ.
   Full step-by-step solution

   Step 1: Start with the left side: (sin⁶θ - cos⁶θ) / (sin²θ - cos²θ). Step 2: Recognize the numerator as a difference of cubes: a³ - b³ = (a - b)(a² + ab + b²). Here a = sin²θ, b = cos²θ. So sin⁶θ - cos⁶θ = (sin²θ - cos²θ)(sin⁴θ + sin²θ cos²θ + cos⁴θ). Step 3: Substitute into the left side: [(sin²θ - cos²θ)(sin⁴θ + sin²θ cos²θ + cos⁴θ)] / (sin²θ - cos²θ). Step 4: Cancel the common factor (sin²θ - cos²θ), provided sin²θ ≠ cos²θ. Step 5: This leaves sin⁴θ + sin²θ cos²θ + cos⁴θ. Step 6: Rewrite sin⁴θ + cos⁴θ as (sin²θ + cos²θ)² - 2 sin²θ cos²θ = 1 - 2 sin²θ cos²θ. Step 7: So the expression becomes (1 - 2 sin²θ cos²θ) + sin²θ cos²θ = 1 - sin²θ cos²θ. Step 8: This simplifies to 1 - sin²θ cos²θ, which equals the right side. Thus the identity is verified.

4. Emma is analyzing a right triangle inscribed in the unit circle on a coordinate plane. The terminal side of angle θ intersects the unit circle at point P with coordinates (x, y). Using the visual representation of the triangle and the circle, verify the identity: (1 - sin²θ) / cosθ = cosθ. Answer: cosθ = cosθ (identity verified) Solution: On the unit circle, the point P has coordinates (cosθ, sinθ). So x = cosθ and y = sinθ. The fundamental Pythagorean identity for the unit circle is x² + y² = 1, which gives cos²θ + sin²θ = 1.
   Full step-by-step solution

   Step 1: On the unit circle, the point P has coordinates (cosθ, sinθ). So x = cosθ and y = sinθ. Step 2: The fundamental Pythagorean identity for the unit circle is x² + y² = 1, which gives cos²θ + sin²θ = 1. Step 3: From cos²θ + sin²θ = 1, rearrange to isolate cos²θ: cos²θ = 1 - sin²θ. Step 4: The left side of the given identity is (1 - sin²θ) / cosθ. Substitute 1 - sin²θ with cos²θ from Step 3. Step 5: This gives (cos²θ) / cosθ. Step 6: Simplify the fraction: cos²θ / cosθ = cosθ (since cos²θ = cosθ * cosθ, dividing by cosθ leaves cosθ). Step 7: The left side simplifies to cosθ, which equals the right side. The identity is verified. The answer is cosθ = cosθ (identity verified).

5. Verify: (sec⁴θ - tan⁴θ) / (sec²θ + tan²θ) = 1. Answer: 1 Solution: Start with the left side: (sec⁴θ - tan⁴θ) / (sec²θ + tan²θ). Factor the numerator as a difference of squares: sec⁴θ - tan⁴θ = (sec²θ - tan²θ)(sec²θ + tan²θ).
   Full step-by-step solution

   Start with the left side: (sec⁴θ - tan⁴θ) / (sec²θ + tan²θ). Factor the numerator as a difference of squares: sec⁴θ - tan⁴θ = (sec²θ - tan²θ)(sec²θ + tan²θ). Cancel the common factor (sec²θ + tan²θ) in numerator and denominator: (sec²θ - tan²θ)(sec²θ + tan²θ) / (sec²θ + tan²θ) = sec²θ - tan²θ. Use the Pythagorean identity: sec²θ = 1 + tan²θ, so sec²θ - tan²θ = 1. Thus the left side simplifies to 1, which equals the right side. The identity is verified.

6. Verify: (sec⁷θ - tan⁷θ) / (secθ - tanθ) = sec⁶θ + sec⁵θ tanθ + sec⁴θ tan²θ + sec³θ tan³θ + sec²θ tan⁴θ + secθ tan⁵θ + tan⁶θ. Answer: Verified identity Solution: Start with the left side: (sec⁷θ - tan⁷θ) / (secθ - tanθ). Recognize the numerator as a difference of seventh powers. For any a and b, a⁷ - b⁷ = (a - b)(a⁶ + a⁵b + a⁴b² + a³b³ + a²b⁴ + ab⁵ + b⁶).
   Full step-by-step solution

   Step 1: Start with the left side: (sec⁷θ - tan⁷θ) / (secθ - tanθ). Step 2: Recognize the numerator as a difference of seventh powers. For any a and b, a⁷ - b⁷ = (a - b)(a⁶ + a⁵b + a⁴b² + a³b³ + a²b⁴ + ab⁵ + b⁶). Here a = secθ, b = tanθ. Step 3: Apply the factorization: sec⁷θ - tan⁷θ = (secθ - tanθ)(sec⁶θ + sec⁵θ tanθ + sec⁴θ tan²θ + sec³θ tan³θ + sec²θ tan⁴θ + secθ tan⁵θ + tan⁶θ). Step 4: Substitute into the left side: [(secθ - tanθ)(sec⁶θ + sec⁵θ tanθ + sec⁴θ tan²θ + sec³θ tan³θ + sec²θ tan⁴θ + secθ tan⁵θ + tan⁶θ)] / (secθ - tanθ). Step 5: Cancel the common factor (secθ - tanθ), provided secθ ≠ tanθ (i.e., θ not equal to π/4 + kπ). Step 6: The result is sec⁶θ + sec⁵θ tanθ + sec⁴θ tan²θ + sec³θ tan³θ + sec²θ tan⁴θ + secθ tan⁵θ + tan⁶θ, which equals the right side. Thus the identity is verified.

7. Liam is designing a suspension bridge where the main cable follows a parabolic curve. The cable's shape is modeled by the function f(x) = 2x² - 8x + 6. To ensure structural integrity, Liam needs to find the points where the cable is perfectly horizontal. At what x-coordinates does the cable have horizontal tangents? Answer: x = 2 Solution: To find where the cable has horizontal tangents, we need to find where the slope of the curve is zero. The slope of the curve at any point is given by the derivative of the function. f(x) = 2x² - 8x + 6 The derivative of 2x² is 4x The derivative of -8x is -8 The derivative of 6 is 0 So f'(x) =…
   Full step-by-step solution

   To find where the cable has horizontal tangents, we need to find where the slope of the curve is zero. The slope of the curve at any point is given by the derivative of the function. Step 1: Write the function f(x) = 2x² - 8x + 6 Step 2: Find the derivative f'(x) The derivative of 2x² is 4x The derivative of -8x is -8 The derivative of 6 is 0 So f'(x) = 4x - 8 Step 3: Set the derivative equal to zero to find where the slope is horizontal 4x - 8 = 0 Step 4: Solve for x 4x = 8 x = 8/4 x = 2 Step 5: Interpret the result The cable has a horizontal tangent at x = 2. ANSWER: x = 2