Sum Difference Formulas

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Sum Difference Formulas · Grade 11 · Worksheet 1

1. cos(285°) = ? Answer: (√6 - √2)/4 Solution: Express 285° as a sum of two special angles: 285° = 225° + 60°. Find exact values: cos(225°) = -√2/2, sin(225°) = -√2/2, cos(60°) = 1/2, sin(60°) = √3/2.
   Full step-by-step solution

   Step 1: Express 285° as a sum of two special angles: 285° = 225° + 60°. Step 2: Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB, with A = 225°, B = 60°. Step 3: Find exact values: cos(225°) = -√2/2, sin(225°) = -√2/2, cos(60°) = 1/2, sin(60°) = √3/2. Step 4: Substitute: cos(285°) = (-√2/2)(1/2) - (-√2/2)(√3/2) = -√2/4 + (√2·√3)/4 = -√2/4 + √6/4 = (√6 - √2)/4. The exact value of cos(285°) is (√6 - √2)/4.

2. cos(165°) = ? Answer: -√6/4 - √2/4 Solution: Express 165° as a sum of angles with known cosine and sine values: 165° = 120° + 45° Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB Substitute A = 120° and B = 45°: cos(165°) = cos(120°)cos(45°) - sin(120°)sin(45°) Use exact values: cos(120°) = -1/2, cos(45°) = √2/2, sin(120°) =…
   Full step-by-step solution

   Step 1: Express 165° as a sum of angles with known cosine and sine values: 165° = 120° + 45° Step 2: Apply the cosine sum formula: cos(A+B) = cosA cosB - sinA sinB Step 3: Substitute A = 120° and B = 45°: cos(165°) = cos(120°)cos(45°) - sin(120°)sin(45°) Step 4: Use exact values: cos(120°) = -1/2, cos(45°) = √2/2, sin(120°) = √3/2, sin(45°) = √2/2 Step 5: Substitute the values: cos(165°) = (-1/2)(√2/2) - (√3/2)(√2/2) Step 6: Simplify: cos(165°) = -√2/4 - √6/4 Step 7: Combine terms: cos(165°) = -√6/4 - √2/4 The answer is -√6/4 - √2/4.

3. Aroha is a marine biologist studying the depth of a tidal channel. She models the depth of the water, in meters, at time t hours after midnight using the function D(t) = 12sin(πt/6) + 5cos(πt/6). To predict the maximum depth for a safe navigation report, Aroha needs to rewrite this expression in the form Rsin(πt/6 + α). Determine the exact value of the amplitude R. Answer: 13 Solution: Write the target form: Rsin(πt/6 + α) = R[sin(πt/6)cos(α) + cos(πt/6)sin(α)] = Rcos(α) sin(πt/6) + Rsin(α) cos(πt/6). Compare with D(t) = 12sin(πt/6) + 5cos(πt/6). This gives: Rcos(α) = 12 and Rsin(α) = 5.
   Full step-by-step solution

   Step 1: Write the target form: Rsin(πt/6 + α) = R[sin(πt/6)cos(α) + cos(πt/6)sin(α)] = Rcos(α) sin(πt/6) + Rsin(α) cos(πt/6). Step 2: Compare with D(t) = 12sin(πt/6) + 5cos(πt/6). This gives: Rcos(α) = 12 and Rsin(α) = 5. Step 3: Square both equations: R²cos²(α) = 144 and R²sin²(α) = 25. Step 4: Add the equations: R²(cos²(α) + sin²(α)) = 144 + 25 = 169. Step 5: Since cos²(α) + sin²(α) = 1, we have R² = 169, so R = 13 (positive amplitude). The answer is 13.

4. tan(165°) = ? Answer: -(2 - √3) Solution: Express 165° as a sum: 165° = 120° + 45°. Use the tangent sum formula: tan(A+B) = (tan A + tan B) / (1 - tan A tan B). tan(120°) = tan(180° - 60°) = -tan(60°) = -√3.
   Full step-by-step solution

   Step 1: Express 165° as a sum: 165° = 120° + 45°. Step 2: Use the tangent sum formula: tan(A+B) = (tan A + tan B) / (1 - tan A tan B). Step 3: tan(120°) = tan(180° - 60°) = -tan(60°) = -√3. Step 4: tan(45°) = 1. Step 5: Substitute: tan(165°) = ( -√3 + 1 ) / ( 1 - (-√3)(1) ) = (1 - √3) / (1 + √3). Step 6: Rationalize the denominator: Multiply numerator and denominator by (1 - √3): (1 - √3)(1 - √3) / ((1 + √3)(1 - √3)) = (1 - 2√3 + 3) / (1 - 3) = (4 - 2√3) / (-2) = -2 + √3. Step 7: Simplify: tan(165°) = -(2 - √3). The exact value is -(2 - √3).

5. cos(135°)cos(45°) + sin(135°)sin(45°) = ? Answer: 0 Solution: Recognize the pattern cos(A)cos(B) + sin(A)sin(B) = cos(A-B) Identify A = 135° and B = 45° Apply the formula: cos(135°-45°) = cos(90°) Evaluate cos(90°) = 0 Therefore, the expression equals 0
   Full step-by-step solution

   Step 1: Recognize the pattern cos(A)cos(B) + sin(A)sin(B) = cos(A-B) Step 2: Identify A = 135° and B = 45° Step 3: Apply the formula: cos(135°-45°) = cos(90°) Step 4: Evaluate cos(90°) = 0 Step 5: Therefore, the expression equals 0

6. Aroha is a marine biologist studying the motion of a buoy in the ocean. The buoy's vertical displacement from sea level (in meters) over time (in seconds) is modeled by the function h(t) = 9 sin(2t) + 12 cos(2t). Using sum and difference formulas, rewrite this expression in the form R sin(2t + α) to find the amplitude of the buoy's motion. What is the exact amplitude? Answer: 15 Solution: We want to write h(t) = 9 sin(2t) + 12 cos(2t) in the form R sin(2t + α). Using the sine sum formula: R sin(2t + α) = R sin(2t) cos(α) + R cos(2t) sin(α). Compare coefficients with 9 sin(2t) + 12 cos(2t).
   Full step-by-step solution

   Step 1: We want to write h(t) = 9 sin(2t) + 12 cos(2t) in the form R sin(2t + α). Step 2: Using the sine sum formula: R sin(2t + α) = R sin(2t) cos(α) + R cos(2t) sin(α). Step 3: Compare coefficients with 9 sin(2t) + 12 cos(2t). This gives: R cos(α) = 9 R sin(α) = 12 Step 4: Square both equations and add: R^2 cos^2(α) + R^2 sin^2(α) = 9^2 + 12^2 R^2 (cos^2(α) + sin^2(α)) = 81 + 144 R^2 = 225 R = 15 (since amplitude is positive) Step 5: The amplitude is R = 15 meters. The answer is 15.

7. Liam is designing a suspension bridge where the main cable follows a sinusoidal pattern. The cable's height above the road surface is modeled by h(x) = 25sin(0.1x) + 30, where x is the horizontal distance from the left tower in meters. To analyze stress points, Liam needs to find the exact height of the cable at x = 15 meters using the sum formula for sine. What is the exact height of the cable at this point? Answer: 25√2/2 + 30 Solution: The sum formula for sine states that sin(A+B) = sinAcosB + cosAsinB.
   Full step-by-step solution

   The sum formula for sine states that sin(A+B) = sinAcosB + cosAsinB. This identity is particularly useful when you need to find exact values of trigonometric functions for angles that can be expressed as sums of special angles like 30°, 45°, 60°, or 90°. In engineering applications, these exact values are often preferred over decimal approximations for precise calculations in structural analysis.

8. Aroha is studying the reflection of light off a surface in her physics class. She needs to find the exact value of cos(105°) to calculate the angle of reflection. Using sum and difference formulas, determine the exact value of cos(105°). Answer: (sqrt(2) - sqrt(6))/4 Solution: Express 105° as a sum of two special angles: 105° = 60° + 45° Apply the cosine sum formula: cos(A + B) = cosA cosB - sinA sinB Substitute A = 60° and B = 45°: cos(105°) = cos(60°)cos(45°) - sin(60°)sin(45°) Use exact values: cos(60°) = 1/2, cos(45°) = sqrt(2)/2, sin(60°) = sqrt(3)/2, 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 the values: cos(105°) = (1/2)(sqrt(2)/2) - (sqrt(3)/2)(sqrt(2)/2) Step 6: Simplify each term: cos(105°) = sqrt(2)/4 - sqrt(6)/4 Step 7: Combine the terms: cos(105°) = (sqrt(2) - sqrt(6))/4 The answer is (sqrt(2) - sqrt(6))/4.