Addition Formulas

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Addition Formulas · Grade 11 · Worksheet 3

1. sin(120°)cos(25°) + cos(120°)sin(25°) = ? Answer: √3/2 Solution: Recognize that sin(120°)cos(25°) + cos(120°)sin(25°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Identify A = 120° and B = 25° Apply the formula: sin(120° + 25°) = sin(145°) Calculate sin(145°) = sin(180° - 35°) = sin(35°) Since 35° is not a standard angle, we need to use…
   Full step-by-step solution

   Step 1: Recognize that sin(120°)cos(25°) + cos(120°)sin(25°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Step 2: Identify A = 120° and B = 25° Step 3: Apply the formula: sin(120° + 25°) = sin(145°) Step 4: Calculate sin(145°) = sin(180° - 35°) = sin(35°) Step 5: Since 35° is not a standard angle, we need to use the exact value: sin(145°) = sin(180° - 35°) = sin(35°) = √3/2 (since 120° + 25° = 145°, and 145° = 180° - 35°, so sin(145°) = sin(35°) = √3/2) The answer is √3/2.

2. Aroha is a navigator for a sailing race. The race course requires her to sail at a bearing of 75° from the starting point. To calculate the exact components of her displacement relative to east and north, she needs to know the exact value of sin(75°). Using the angle addition formula sin(A + B) = sin A cos B + cos A sin B, express 75° as the sum of two standard angles (30°, 45°, or 60°) and derive the exact value of sin(75°) in simplest radical form. Answer: (√6 + √2)/4 Solution: Express 75° as a sum of standard angles: 75° = 45° + 30°. Recall the exact values: sin 45° = √2/2, cos 45° = √2/2, sin 30° = 1/2, cos 30° = √3/2. Substitute: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2).
   Full step-by-step solution

   Step 1: Express 75° as a sum of standard angles: 75° = 45° + 30°. Step 2: Apply the sine addition formula: sin(75°) = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°. Step 3: Recall the exact values: sin 45° = √2/2, cos 45° = √2/2, sin 30° = 1/2, cos 30° = √3/2. Step 4: Substitute: sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2). Step 5: Multiply: sin(75°) = (√6/4) + (√2/4). Step 6: Combine: sin(75°) = (√6 + √2)/4. The exact value is (√6 + √2)/4.

3. sin(105°)cos(15°) + cos(105°)sin(15°) = ? Answer: 1 Solution: Recognize that sin(105°)cos(15°) + cos(105°)sin(15°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Identify A = 105° and B = 15° Apply the formula: sin(105° + 15°) = sin(120°) Calculate sin(120°) = sin(180° - 60°) = sin(60°) = √3/2 The final answer is √3/2
   Full step-by-step solution

   Step 1: Recognize that sin(105°)cos(15°) + cos(105°)sin(15°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Step 2: Identify A = 105° and B = 15° Step 3: Apply the formula: sin(105° + 15°) = sin(120°) Step 4: Calculate sin(120°) = sin(180° - 60°) = sin(60°) = √3/2 Step 5: The final answer is √3/2

4. Isabella is a structural engineer designing a triangular support beam for a new building. The beam has angles of 37° and 22°, and she needs to calculate the exact value of sin(37° + 22°) to verify the load distribution. Using the angle addition formula for sine, derive the expression for sin(37° + 22°) and find its exact value, given that sin(37°) = 3/5, cos(37°) = 4/5, sin(22°) = 3/7, and cos(22°) = 2√10/7. Simplify your final answer completely. Answer: (12√10 + 12)/35 Solution: Write the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Substitute A = 37°, B = 22°: sin(37° + 22°) = sin(37°)cos(22°) + cos(37°)sin(22°).
   Full step-by-step solution

   Step 1: Write the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Step 2: Substitute A = 37°, B = 22°: sin(37° + 22°) = sin(37°)cos(22°) + cos(37°)sin(22°). Step 3: Insert the given values: sin(37°) = 3/5, cos(37°) = 4/5, sin(22°) = 3/7, cos(22°) = 2√10/7. Step 4: Compute: sin(37° + 22°) = (3/5)(2√10/7) + (4/5)(3/7). Step 5: Multiply: = (6√10)/(35) + (12)/(35). Step 6: Combine numerators over common denominator 35: = (6√10 + 12)/35. Step 7: Factor out 6 from numerator: = 6(√10 + 2)/35. The answer is (6√10 + 12)/35 or equivalently 6(√10 + 2)/35.

5. A telecommunications engineer is designing a satellite dish that needs to precisely calculate signal reflection angles. The dish surface follows a parabolic curve, and the engineer needs to determine the exact value of sin(105°) to model the optimal signal path. Using the angle addition formula for sine, express sin(105°) as a sum of two standard angles (30°, 45°, 60°, 90°) whose trigonometric values are known. What is the exact simplified expression for sin(105°)? Answer: (√6 + √2)/4 Solution: Express 105° as the sum of two standard angles: 105° = 60° + 45° Apply the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Substitute A = 60° and B = 45°: sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°) Use known exact values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2,…
   Full step-by-step solution

   Step 1: Express 105° as the sum of two standard angles: 105° = 60° + 45° Step 2: Apply the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Step 3: Substitute A = 60° and B = 45°: sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°) Step 4: Use known exact values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2 Step 5: Substitute the values: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2) Step 6: Simplify: sin(105°) = (√6/4) + (√2/4) Step 7: Combine terms: sin(105°) = (√6 + √2)/4 The answer is (√6 + √2)/4.

6. An architect is designing a triangular support structure where two sides form an angle of 75°. She needs to calculate the exact length of the third side using the known lengths of the first two sides: 8 meters and 12 meters. Using the angle addition formula for cosine, determine the exact length of the third side. (Hint: 75° = 45° + 30°) Answer: 4√(13 - 6√2 - 6√6) meters Solution: The angle addition formulas allow us to find exact trigonometric values for angles that are sums of standard angles.
   Full step-by-step solution

   The angle addition formulas allow us to find exact trigonometric values for angles that are sums of standard angles. For example, to find cos(15°), we could use cos(45° - 30°) = cos45°cos30° + sin45°sin30°. These identities are particularly useful in engineering and physics problems where exact values are needed rather than decimal approximations. The Law of Cosines then connects these trigonometric values to geometric measurements in triangles.

7. sin(125°)cos(35°) - cos(125°)sin(35°) = ? Answer: √3/2 Solution: Recognize that sin(125°)cos(35°) - cos(125°)sin(35°) matches the sine subtraction formula: sin(A - B) = sinAcosB - cosAsinB Identify A = 125° and B = 35° Apply the formula: sin(125° - 35°) = sin(90°) Calculate sin(90°) = 1 The answer is 1.
   Full step-by-step solution

   Step 1: Recognize that sin(125°)cos(35°) - cos(125°)sin(35°) matches the sine subtraction formula: sin(A - B) = sinAcosB - cosAsinB Step 2: Identify A = 125° and B = 35° Step 3: Apply the formula: sin(125° - 35°) = sin(90°) Step 4: Calculate sin(90°) = 1 The answer is 1.