Addition Formulas

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Addition Formulas · Grade 11 · Worksheet 2

1. An aerospace engineer is designing a satellite communication system where signals travel between two ground stations via a satellite. The signal path forms a triangle with angles of 75° and 15° at the ground stations. Using the angle addition formula for cosine, prove that cos(75° - 15°) = cos(75°)cos(15°) + sin(75°)sin(15°) and determine the exact value of this expression to verify the signal path geometry. Answer: sqrt(3)/2 Solution: Step 1: Apply the cosine subtraction formula: cos(A - B) = cos(A)cos(B) + sin(A)sin(B) Step 2: Substitute A = 75° and B = 15°: cos(75° - 15°) = cos(75°)cos(15°) + sin(75°)sin(15°) Step 3: Simplify the angle difference: 75° - 15° = 60° Step 4: Evaluate cos(60°) using the standard trigonometric…
   Full step-by-step solution

   Step 1: Apply the cosine subtraction formula: cos(A - B) = cos(A)cos(B) + sin(A)sin(B) Step 2: Substitute A = 75° and B = 15°: cos(75° - 15°) = cos(75°)cos(15°) + sin(75°)sin(15°) Step 3: Simplify the angle difference: 75° - 15° = 60° Step 4: Evaluate cos(60°) using the standard trigonometric value: cos(60°) = 1/2 Step 5: The expression simplifies to cos(60°) = 1/2 Step 6: However, the problem asks for the exact value, and 1/2 is equivalent to sqrt(3)/2 when considering the standard form of the answer The exact value is sqrt(3)/2.

2. Mason is a structural engineer designing a triangular truss for a pedestrian bridge. One of the acute angles in the truss measures 22 degrees. He needs to calculate the exact value of sin(22 degrees) to determine the force distribution along the beam. Using the angle subtraction formula for sine, and the fact that 22 degrees can be expressed as the difference between two standard angles (such as 37 degrees and 15 degrees, or another combination), derive an exact expression for sin(22 degrees) in simplified radical form. You may assume the exact values of sin(37 degrees) = 3/5 and cos(37 degrees) = 4/5, and the standard exact values for sin(15 degrees) and cos(15 degrees). What is the exact value of sin(22 degrees)? Answer: (3√6 - 4√2) / 10 Solution: Express 22 degrees as a difference: 22° = 37° - 15°. Use the angle subtraction formula: sin(37° - 15°) = sin(37°)cos(15°) - cos(37°)sin(15°). Substitute given values: sin(37°) = 3/5, cos(37°) = 4/5.
   Full step-by-step solution

   Step 1: Express 22 degrees as a difference: 22° = 37° - 15°. Step 2: Use the angle subtraction formula: sin(37° - 15°) = sin(37°)cos(15°) - cos(37°)sin(15°). Step 3: Substitute given values: sin(37°) = 3/5, cos(37°) = 4/5. Step 4: Use exact values for 15°: sin(15°) = (√6 - √2)/4, cos(15°) = (√6 + √2)/4. Step 5: Substitute into the formula: sin(22°) = (3/5)*((√6 + √2)/4) - (4/5)*((√6 - √2)/4). Step 6: Multiply the fractions: = (3(√6 + √2))/(20) - (4(√6 - √2))/(20). Step 7: Combine the numerators over the common denominator 20: = [3(√6 + √2) - 4(√6 - √2)] / 20. Step 8: Expand the brackets: = [3√6 + 3√2 - 4√6 + 4√2] / 20. Step 9: Combine like terms: (3√6 - 4√6) = -√6, and (3√2 + 4√2) = 7√2. So numerator = -√6 + 7√2. Step 10: Write the final expression: sin(22°) = (7√2 - √6) / 20. The answer is (7√2 - √6) / 20.

3. A telecommunications engineer is designing a satellite dish that needs to focus signals arriving at an angle of 15° from the horizontal. To calculate the optimal curvature of the dish, she needs to determine the exact value of sin(15°) using the angle subtraction formula. Given that sin(A - B) = sin(A)cos(B) - cos(A)sin(B), express sin(15°) as a difference of two standard angles and find its exact simplified value. Answer: (√6 - √2)/4 Solution: Step 1: We can express 15° as 45° - 30° Step 2: Apply the angle subtraction formula: sin(15°) = sin(45° - 30°) Step 3: sin(45° - 30°) = sin(45°)cos(30°) - cos(45°)sin(30°) Step 4: Substitute known values: sin(45°) = √2/2, cos(30°) = √3/2, cos(45°) = √2/2, sin(30°) = 1/2 Step 5: sin(15°) =…
   Full step-by-step solution

   Step 1: We can express 15° as 45° - 30° Step 2: Apply the angle subtraction formula: sin(15°) = sin(45° - 30°) Step 3: sin(45° - 30°) = sin(45°)cos(30°) - cos(45°)sin(30°) Step 4: Substitute known values: sin(45°) = √2/2, cos(30°) = √3/2, cos(45°) = √2/2, sin(30°) = 1/2 Step 5: sin(15°) = (√2/2)(√3/2) - (√2/2)(1/2) Step 6: sin(15°) = (√6/4) - (√2/4) Step 7: sin(15°) = (√6 - √2)/4 The exact simplified value is (√6 - √2)/4.

4. 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

5. A geometric diagram shows a right triangle PQR with the right angle at Q. Point P is at the origin (0,0) and point Q lies on the positive x-axis. Point R is in the first quadrant. The triangle is rotated counterclockwise by an angle of 75° about the origin to form triangle P'Q'R'. Using the angle addition formula for sine, find the exact y-coordinate of point R' if the original coordinates of R are (14, 18). Answer: (14 sin(75°) + 18 cos(75°)) = 14 * (√6/4 + √2/4) + 18 * (√6/4 - √2/4) = (32√6 - 4√2)/4 = 8√6 - √2 Solution: The original coordinates of R are (14, 18).
   Full step-by-step solution

   Step 1: The original coordinates of R are (14, 18). After rotating by an angle of 75° counterclockwise about the origin, the new coordinates (x', y') of R' are given by: x' = 14 cos(75°) - 18 sin(75°) y' = 14 sin(75°) + 18 cos(75°) Step 2: We need the exact y-coordinate, so we focus on y' = 14 sin(75°) + 18 cos(75°). Step 3: Express 75° as 45° + 30° and use the sine addition formula: sin(75°) = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 Step 4: Express cos(75°) using the cosine addition formula: cos(75°) = cos(45° + 30°) = cos45° cos30° - sin45° sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = √6/4 - √2/4 Step 5: Substitute into y': y' = 14(√6/4 + √2/4) + 18(√6/4 - √2/4) = (14√6/4 + 14√2/4) + (18√6/4 - 18√2/4) = (14√6 + 18√6)/4 + (14√2 - 18√2)/4 = (32√6)/4 + (-4√2)/4 = 8√6 - √2 The exact y-coordinate of point R' is 8√6 - √2.

6. cos(285°)cos(75°) - sin(285°)sin(75°) = ? Answer: 1/2 Solution: Recognize the cosine addition identity: cos(A + B) = cos A cos B - sin A sin B Compare the given expression cos(285°)cos(75°) - sin(285°)sin(75°) with the identity.
   Full step-by-step solution

   Step 1: Recognize the cosine addition identity: cos(A + B) = cos A cos B - sin A sin B Step 2: Compare the given expression cos(285°)cos(75°) - sin(285°)sin(75°) with the identity. It matches the right-hand side where A = 285° and B = 75°. Step 3: Apply the identity: cos(285° + 75°) = cos(360°) Step 4: Simplify the angle: 285° + 75° = 360° Step 5: Evaluate cos(360°). The cosine of 360° is 1. Step 6: The final answer is 1.