Sum Difference Formulas

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Answer Key & Explanations

Sum Difference Formulas · Grade 11 · Worksheet 2

1. Olivia is studying the path of a projectile in her physics class. The height of the projectile at time t seconds is given by h(t) = 3 sin(5t) + 4 cos(5t). Using the sum formula for sine, rewrite h(t) in the form R sin(5t + α) and determine the exact amplitude R of the projectile's motion. Answer: 5 Solution: We have h(t) = 3 sin(5t) + 4 cos(5t). We want to write it as R sin(5t + α). Using the identity: a sin θ + b cos θ = R sin(θ + α), where R = sqrt(a² + b²) and α satisfies cos α = a/R and sin α = b/R.
   Full step-by-step solution

   Step 1: We have h(t) = 3 sin(5t) + 4 cos(5t). We want to write it as R sin(5t + α). Step 2: Using the identity: a sin θ + b cos θ = R sin(θ + α), where R = sqrt(a² + b²) and α satisfies cos α = a/R and sin α = b/R. Step 3: Here a = 3 and b = 4. Compute R = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5. Step 4: So R = 5, which is the amplitude of the motion. Step 5: Therefore, h(t) = 5 sin(5t + α), where α = arctan(4/3) (but not needed for amplitude). The answer is 5.

2. Liam is designing a suspension bridge and needs to calculate the exact height of the main cable at a point 30 meters from the center of the bridge. The cable follows a parabolic curve described by the equation y = 25 - (x²/36), where y is the height in meters and x is the horizontal distance from the center. Using trigonometric identities, determine the exact height of the cable when x = 30 meters. Answer: 0 Solution: y = 25 - (x² / 36) - x is the horizontal distance from the center of the bridge We need the height when x = 30 meters.
   Full step-by-step solution

   Let's go step by step. We are given the equation of the cable: y = 25 - (x² / 36) Here: - y is the height in meters - x is the horizontal distance from the center of the bridge We need the height when x = 30 meters. --- **Step 1: Substitute x = 30 into the equation** y = 25 - (30² / 36) --- **Step 2: Calculate 30²** 30² = 900 So: y = 25 - (900 / 36) --- **Step 3: Simplify 900 / 36** Divide 900 by 36: 900 ÷ 36 = 25 So: y = 25 - 25 --- **Step 4: Final subtraction** y = 0 --- **Conclusion:** The height of the cable at x = 30 meters is exactly 0 meters. This means that at 30 meters from the center, the cable is at the same level as the bridge deck (assuming y=0 corresponds to deck height at that point).

3. cos(105°) = cos(45° + 60°) = ? Answer: (√2 - √6)/4 Solution: Step 1: Apply the cosine sum formula: cos(45° + 60°) = cos45°cos60° - sin45°sin60° Step 2: Substitute exact values: cos45° = √2/2, cos60° = 1/2, sin45° = √2/2, sin60° = √3/2 Step 3: Multiply: (√2/2)(1/2) - (√2/2)(√3/2) = √2/4 - √6/4 Step 4: Combine terms: (√2 - √6)/4 The answer is (√2 - √6)/4.
   Full step-by-step solution

   Step 1: Apply the cosine sum formula: cos(45° + 60°) = cos45°cos60° - sin45°sin60° Step 2: Substitute exact values: cos45° = √2/2, cos60° = 1/2, sin45° = √2/2, sin60° = √3/2 Step 3: Multiply: (√2/2)(1/2) - (√2/2)(√3/2) = √2/4 - √6/4 Step 4: Combine terms: (√2 - √6)/4 The answer is (√2 - √6)/4.

4. Liam is an engineer designing a solar panel array. The efficiency of a panel depends on the angle of sunlight, and for a particular calculation, he needs the exact value of cos(105°). Using the sum and difference formulas for cosine, determine the exact value of cos(105°) by expressing 105° as the sum of two special angles (for example, 60° and 45°). Answer: (sqrt(2) - sqrt(6))/4 Solution: Express 105° as a sum of two special angles: 105° = 60° + 45°. Substitute A = 60° and B = 45°: cos(105°) = cos(60°)cos(45°) - sin(60°)sin(45°).
   Full step-by-step solution

   Step 1: Express 105° as a sum of two special angles: 105° = 60° + 45°. Step 2: Apply the cosine sum formula: cos(A + B) = cosA cosB - sinA sinB. Step 3: Substitute A = 60° and B = 45°: cos(105°) = cos(60°)cos(45°) - sin(60°)sin(45°). Step 4: Use exact values: cos(60°) = 1/2, cos(45°) = sqrt(2)/2, sin(60°) = sqrt(3)/2, sin(45°) = sqrt(2)/2. Step 5: Substitute: cos(105°) = (1/2)(sqrt(2)/2) - (sqrt(3)/2)(sqrt(2)/2). Step 6: Simplify: cos(105°) = sqrt(2)/4 - sqrt(6)/4. Step 7: Combine: cos(105°) = (sqrt(2) - sqrt(6))/4. The answer is (sqrt(2) - sqrt(6))/4.

5. Emma is analyzing sound wave interference patterns in her physics lab. She has two sound waves represented by the functions f(t) = 3sin(2t) and g(t) = 4cos(2t), where t is time in seconds. Using trigonometric identities, determine the amplitude of the combined wave h(t) = f(t) + g(t) when expressed in the form Rsin(2t + α). Answer: 5 Solution: Write the combined wave function: h(t) = 3sin(2t) + 4cos(2t) We want to express this in the form Rsin(2t + α) = R[sin(2t)cos(α) + cos(2t)sin(α)] Compare coefficients: Rcos(α) = 3 and Rsin(α) = 4 Square both equations and add them: (Rcos(α))² + (Rsin(α))² = 3² + 4² R²(cos²(α) + sin²(α)) = 9 + 16…
   Full step-by-step solution

   Step 1: Write the combined wave function: h(t) = 3sin(2t) + 4cos(2t) Step 2: We want to express this in the form Rsin(2t + α) = R[sin(2t)cos(α) + cos(2t)sin(α)] Step 3: Compare coefficients: Rcos(α) = 3 and Rsin(α) = 4 Step 4: Square both equations and add them: (Rcos(α))² + (Rsin(α))² = 3² + 4² Step 5: R²(cos²(α) + sin²(α)) = 9 + 16 Step 6: Since cos²(α) + sin²(α) = 1, we get R² = 25 Step 7: Therefore, R = 5 Step 8: The amplitude of the combined wave is 5 Step 9: The answer is 5

6. sin(165°) = ? Answer: (√6 - √2)/4 Solution: Express 165° as a sum or difference of known angles. A good choice is 165° = 120° + 45°. Substitute A = 120° and B = 45°: sin(165°) = sin(120°)cos(45°) + cos(120°)sin(45°).
   Full step-by-step solution

   Step 1: Express 165° as a sum or difference of known angles. A good choice is 165° = 120° + 45°. Step 2: Apply the sine sum formula: sin(A + B) = sin A cos B + cos A sin B. Step 3: Substitute A = 120° and B = 45°: sin(165°) = sin(120°)cos(45°) + cos(120°)sin(45°). Step 4: Find the exact values: sin(120°) = √3/2, cos(45°) = √2/2, cos(120°) = -1/2, sin(45°) = √2/2. Step 5: Substitute the values: sin(165°) = (√3/2)(√2/2) + (-1/2)(√2/2) = (√6/4) - (√2/4). Step 6: Combine the terms: (√6/4) - (√2/4) = (√6 - √2)/4. The exact value of sin(165°) is (√6 - √2)/4.

7. Liam is designing a suspension bridge and needs to calculate the exact height of the main cable at a specific point. The cable follows the curve y = 50sin(πx/200) + 50, where x is the horizontal distance from the left tower in meters. Using the sum-to-product identities, Liam needs to find the exact value of the cable's height when x = 50 meters. What is the exact height of the cable at this point? Answer: 50 + 25√2 Solution: We are given the cable curve: y = 50 sin(πx/200) + 50. We want the height at x = 50 meters. Substitute x = 50 into the equation.
   Full step-by-step solution

   We are given the cable curve: y = 50 sin(πx/200) + 50. We want the height at x = 50 meters. Step 1: Substitute x = 50 into the equation. y = 50 sin(π * 50 / 200) + 50 Step 2: Simplify the argument of the sine function. π * 50 / 200 = π * (50/200) = π * (1/4) = π/4. So y = 50 sin(π/4) + 50. Step 3: Recall the exact value of sin(π/4). sin(π/4) = √2 / 2. So y = 50 * (√2 / 2) + 50. Step 4: Simplify the multiplication. 50 * (√2 / 2) = (50/2) * √2 = 25√2. Step 5: Add the constant term. y = 25√2 + 50, which is the same as 50 + 25√2. Final answer: 50 + 25√2.