Addition Formulas

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Addition Formulas · Grade 11 · Worksheet 1

1. Hana is a marine biologist studying wave patterns. She models the height of a wave (in meters) over time using the function h(t) = 5 sin(2t + 15°) where t is time in seconds. To analyze the wave's behavior, she needs to rewrite this as a sum of sine and cosine functions. Use the angle addition formula for sine to express h(t) in the form A sin(2t) + B cos(2t), then determine the exact values of A and B. Answer: A = (5√6 + 5√2)/4, B = (5√6 - 5√2)/4 Solution: Use the sine addition formula: sin(A + B) = sin A cos B + cos A sin B. Here A = 2t and B = 15°, so h(t) = 5[sin(2t)cos(15°) + cos(2t)sin(15°)].
   Full step-by-step solution

   Step 1: Use the sine addition formula: sin(A + B) = sin A cos B + cos A sin B. Here A = 2t and B = 15°, so h(t) = 5[sin(2t)cos(15°) + cos(2t)sin(15°)]. Step 2: Find exact values of sin(15°) and cos(15°) using subtraction formulas: sin(15°) = sin(45° - 30°) = sin45°cos30° - cos45°sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6/4) - (√2/4) = (√6 - √2)/4. cos(15°) = cos(45° - 30°) = cos45°cos30° + sin45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6/4) + (√2/4) = (√6 + √2)/4. Step 3: Substitute into h(t): h(t) = 5[sin(2t)((√6 + √2)/4) + cos(2t)((√6 - √2)/4)]. Step 4: Distribute the 5: h(t) = (5(√6 + √2)/4) sin(2t) + (5(√6 - √2)/4) cos(2t). Step 5: Thus A = 5(√6 + √2)/4 and B = 5(√6 - √2)/4. The answer is A = (5√6 + 5√2)/4, B = (5√6 - 5√2)/4.

2. Sophia is an aerospace engineer designing a satellite orbit. She needs to calculate the exact altitude of the satellite at a specific point where the angle from the Earth's center is 105°. Using the angle subtraction formula for sine, derive the exact value of sin(105°) by expressing 105° as the sum of two standard angles (60° and 45°). Show the derivation step by step, then state the exact simplified value. Answer: (√6 + √2)/4 Solution: Express 105° as a sum: 105° = 60° + 45°. Substitute exact values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2. Compute: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2) = (√6/4) + (√2/4).
   Full step-by-step solution

   Step 1: Express 105° as a sum: 105° = 60° + 45°. Step 2: Apply the sine addition formula: sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°). Step 3: Substitute exact values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2. Step 4: Compute: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2) = (√6/4) + (√2/4). Step 5: Combine: sin(105°) = (√6 + √2)/4. The exact value of sin(105°) is (√6 + √2)/4.

3. sin(75°) = ? Answer: (√6 + √2)/4 Solution: A good choice is 45° and 30° because: 45° + 30° = 75° sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°) sin(45°) = √2/2 cos(30°) = √3/2 cos(45°) = √2/2 sin(30°) = 1/2 sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) First term: (√2/2)(√3/2) = (√2 * √3) / (2 * 2) = √6 / 4 Second term:…
   Full step-by-step solution

   We can find sin(75°) using the sum of angles formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Step 1: Choose two angles we know from the unit circle that add to 75°. A good choice is 45° and 30° because: 45° + 30° = 75° Step 2: Write the formula: sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°) Step 3: Substitute known exact values: sin(45°) = √2/2 cos(30°) = √3/2 cos(45°) = √2/2 sin(30°) = 1/2 So: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) Step 4: Multiply the terms: First term: (√2/2)(√3/2) = (√2 * √3) / (2 * 2) = √6 / 4 Second term: (√2/2)(1/2) = (√2 * 1) / (2 * 2) = √2 / 4 Step 5: Add the two terms: sin(75°) = √6/4 + √2/4 Step 6: Since both terms have the same denominator, combine them: sin(75°) = (√6 + √2) / 4 Final answer: (√6 + √2)/4

4. Mason is an astrophysicist analyzing the trajectory of a comet approaching Earth. He determines that the angle between the comet's path and the line from Earth to the Sun is 75 degrees. To calculate the comet's orbital energy, he needs the exact value of cos(75 degrees). Use the cosine subtraction formula, cos(A - B) = cos A cos B + sin A sin B, to express cos(75 degrees) as a combination of known standard angles and find its exact simplified value. Answer: (√6 - √2)/4 Solution: Write 75 degrees as a difference of two standard angles: 75 = 120 - 45, or 75 = 135 - 60. Find exact values: cos(120) = -1/2, cos(45) = √2/2, sin(120) = √3/2, sin(45) = √2/2.
   Full step-by-step solution

   Step 1: Write 75 degrees as a difference of two standard angles: 75 = 120 - 45, or 75 = 135 - 60. We will use 75 = 120 - 45. Step 2: Apply the cosine subtraction formula: cos(120 - 45) = cos(120)cos(45) + sin(120)sin(45). Step 3: Find exact values: cos(120) = -1/2, cos(45) = √2/2, sin(120) = √3/2, sin(45) = √2/2. Step 4: Substitute: cos(75) = (-1/2)(√2/2) + (√3/2)(√2/2) = -√2/4 + √6/4. Step 5: Combine: cos(75) = (√6 - √2)/4. The answer is (√6 - √2)/4.

5. sin(135°)cos(45°) - cos(135°)sin(45°) = ? Answer: 1 Solution: Recall the trigonometric identity for sine of a difference: sin(A - B) = sin A cos B - cos A sin B Compare the given expression with the identity.
   Full step-by-step solution

   Step 1: Recall the trigonometric identity for sine of a difference: sin(A - B) = sin A cos B - cos A sin B Step 2: Compare the given expression with the identity. Given: sin(135°)cos(45°) - cos(135°)sin(45°) matches the right-hand side where A = 135° and B = 45° Step 3: Apply the identity: sin(135°)cos(45°) - cos(135°)sin(45°) = sin(135° - 45°) Step 4: Simplify the angle: 135° - 45° = 90° Step 5: Evaluate sin(90°) = 1 Step 6: The final answer is 1

6. Isabella is a structural engineer analyzing forces in a truss bridge. She encounters a joint where two members meet at an angle of 75 degrees. To calculate the resultant force, she needs to find the exact value of cos(75°). Using the angle subtraction formula for cosine, where cos(A - B) = cos A cos B + sin A sin B, express cos(75°) as a difference of two standard angles (30°, 45°, 60°, 90°) and determine its exact simplified value. Answer: (√6 - √2)/4 Solution: Express 75° as a difference of two standard angles: 75° = 120° - 45°. Use the cosine subtraction formula: cos(A - B) = cos A cos B + sin A sin B, where A = 120° and B = 45°.
   Full step-by-step solution

   Step 1: Express 75° as a difference of two standard angles: 75° = 120° - 45°. Step 2: Use the cosine subtraction formula: cos(A - B) = cos A cos B + sin A sin B, where A = 120° and B = 45°. Step 3: Substitute: cos(120° - 45°) = cos(120°)cos(45°) + sin(120°)sin(45°). Step 4: Recall exact values: cos(120°) = -1/2, cos(45°) = √2/2, sin(120°) = √3/2, sin(45°) = √2/2. Step 5: Substitute these values: cos(75°) = (-1/2)(√2/2) + (√3/2)(√2/2). Step 6: Multiply: cos(75°) = -√2/4 + √6/4. Step 7: Combine: cos(75°) = (√6 - √2)/4. The answer is (√6 - √2)/4.

7. sin(105°)cos(15°) - cos(105°)sin(15°) = ? Answer: 1/2 Solution: Recognize that the expression matches the sine subtraction formula: sin(A - B) = sinAcosB - cosAsinB Identify A = 105° and B = 15° Apply the formula: sin(105° - 15°) = sin(90°) Evaluate sin(90°) = 1 Therefore, sin(105°)cos(15°) - cos(105°)sin(15°) = 1 The answer is 1.
   Full step-by-step solution

   Step 1: Recognize that the expression matches the sine subtraction formula: sin(A - B) = sinAcosB - cosAsinB Step 2: Identify A = 105° and B = 15° Step 3: Apply the formula: sin(105° - 15°) = sin(90°) Step 4: Evaluate sin(90°) = 1 Step 5: Therefore, sin(105°)cos(15°) - cos(105°)sin(15°) = 1 The answer is 1.