Trigonometric Identities

lessonbunny.com

Answer Key & Explanations

Trigonometric Identities · Grade 12 · Worksheet 2

1. Find the exact value of the limit: lim(x→0) (sin(3x) - 3sin(x)) / (x^3) Answer: -4 Solution: Recall the Maclaurin series expansion: sin(x) = x - x^3/6 + x^5/120 - ... Substitute into the numerator: sin(3x) - 3sin(x) = [3x - (3x)^3/6 + ...] - 3[x - x^3/6 + ...] Simplify: sin(3x) - 3sin(x) = 3x - 27x^3/6 + ...
   Full step-by-step solution

   Step 1: Recall the Maclaurin series expansion: sin(x) = x - x^3/6 + x^5/120 - ... Step 2: Substitute into the numerator: sin(3x) - 3sin(x) = [3x - (3x)^3/6 + ...] - 3[x - x^3/6 + ...] Step 3: Simplify: sin(3x) - 3sin(x) = 3x - 27x^3/6 + ... - 3x + 3x^3/6 + ... Step 4: Combine like terms: (3x - 3x) + (-27x^3/6 + 3x^3/6) + ... = -24x^3/6 + ... = -4x^3 + ... Step 5: The expression becomes: (-4x^3 + higher order terms) / x^3 Step 6: As x→0, the limit is -4 The answer is -4.

2. An architect is designing a modern building with a curved roof that follows the path of a trigonometric function. The roof's cross-section is modeled by the equation y = 2sin(x) + 3cos(x). To determine the structural integrity, the architect needs to verify that this function can be rewritten in the form y = Rsin(x + α), where R > 0 and α is a phase shift. Find the values of R and α that satisfy this identity. Answer: R = √13, α = arctan(3/2) Solution: y = 2 sin(x) + 3 cos(x) y = R sin(x + α) sin(x + α) = sin(x) cos(α) + cos(x) sin(α) R sin(x + α) = R cos(α) sin(x) + R sin(α) cos(x) Compare with y = 2 sin(x) + 3 cos(x): For sin(x): R cos(α) = 2 For cos(x): R sin(α) = 3 [R cos(α)]^2 + [R sin(α)]^2 = 2^2 + 3^2 R^2 [cos^2(α) + sin^2(α)] = 4 + 9…
   Full step-by-step solution

   We start with the given function: y = 2 sin(x) + 3 cos(x) We want to write it in the form: y = R sin(x + α) --- **Step 1: Expand R sin(x + α)** Using the sine addition formula: sin(x + α) = sin(x) cos(α) + cos(x) sin(α) So: R sin(x + α) = R cos(α) sin(x) + R sin(α) cos(x) --- **Step 2: Match coefficients** Compare with y = 2 sin(x) + 3 cos(x): For sin(x): R cos(α) = 2 For cos(x): R sin(α) = 3 --- **Step 3: Solve for R** Square both equations and add: [R cos(α)]^2 + [R sin(α)]^2 = 2^2 + 3^2 R^2 [cos^2(α) + sin^2(α)] = 4 + 9 R^2 * 1 = 13 R = √13 (since R > 0) --- **Step 4: Solve for α** From R cos(α) = 2 and R sin(α) = 3: Divide the second by the first: [R sin(α)] / [R cos(α)] = 3/2 tan(α) = 3/2 So α = arctan(3/2) --- **Step 5: Check quadrant** Since R cos(α) = 2 > 0 and R sin(α) = 3 > 0, cos(α) > 0 and sin(α) > 0, so α is in the first quadrant. Thus α = arctan(3/2) is correct without needing to adjust by π. --- **Final answer:** R = √13 α = arctan(3/2)

3. 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.

4. 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.

5. An architect is designing a modern building with a curved roof that follows the path of a trigonometric function. The roof's cross-section is modeled by the equation y = 3sin(x) + 4cos(x). To determine the maximum height of the roof, the architect needs to find the amplitude of this function. What is the amplitude of y = 3sin(x) + 4cos(x)? Answer: 5 Solution: y = 3 sin(x) + 4 cos(x) We want the amplitude of this combined sine-cosine wave. y = a sin(x) + b cos(x) y = R sin(x + φ) where R is the amplitude.
   Full step-by-step solution

   We are given the function: y = 3 sin(x) + 4 cos(x) We want the amplitude of this combined sine-cosine wave. --- **Step 1: Recognize the form** A function of the type y = a sin(x) + b cos(x) can be rewritten as a single sine wave: y = R sin(x + φ) where R is the amplitude. --- **Step 2: Formula for amplitude** For y = a sin(x) + b cos(x), the amplitude R is given by: R = sqrt(a^2 + b^2) Here, a = 3, b = 4. --- **Step 3: Apply the formula** R = sqrt(3^2 + 4^2) R = sqrt(9 + 16) R = sqrt(25) R = 5 --- **Step 4: Conclusion** The amplitude of y = 3 sin(x) + 4 cos(x) is 5. --- **Final answer:** 5

6. Verify: (sec⁴θ - tan⁴θ) / (sec²θ - tan²θ) = sec²θ + tan²θ. Answer: Verified identity 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

   Step 1: Start with the left side: (sec⁴θ - tan⁴θ) / (sec²θ - tan²θ). Step 2: Factor the numerator as a difference of squares: sec⁴θ - tan⁴θ = (sec²θ - tan²θ)(sec²θ + tan²θ). Step 3: Substitute into the left side: [(sec²θ - tan²θ)(sec²θ + tan²θ)] / (sec²θ - tan²θ). Step 4: Cancel the common factor (sec²θ - tan²θ) (provided sec²θ ≠ tan²θ, i.e., θ not a multiple of π/2). Step 5: The result is sec²θ + tan²θ, which equals the right side. Thus the identity is verified.