Addition Formulas

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Addition Formulas · Grade 11 · Worksheet 3

1. Mason is a telecommunications engineer designing a signal reflector that must redirect a beam at an angle of 105°. To program the reflector's orientation, he needs to know the exact value of cos(105°). Using the angle subtraction formula for cosine, express 105° as the sum of two standard angles (45° and 60°), then derive the exact value of cos(105°) in simplified radical form. Answer: (sqrt(2) - sqrt(6))/4 Solution: Express 105° as 45° + 60°. Use the cosine addition formula: cos(105°) = cos(45° + 60°) = cos(45°)cos(60°) - sin(45°)sin(60°).
   Full step-by-step solution

   Step 1: Express 105° as 45° + 60°. Step 2: Use the cosine addition formula: cos(105°) = cos(45° + 60°) = cos(45°)cos(60°) - sin(45°)sin(60°). Step 3: Substitute exact values: cos(45°) = sqrt(2)/2, cos(60°) = 1/2, sin(45°) = sqrt(2)/2, sin(60°) = sqrt(3)/2. Step 4: Compute: cos(105°) = (sqrt(2)/2)(1/2) - (sqrt(2)/2)(sqrt(3)/2) = sqrt(2)/4 - sqrt(6)/4. Step 5: Combine: cos(105°) = (sqrt(2) - sqrt(6))/4. The exact value is (sqrt(2) - sqrt(6))/4.

2. 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.

3. Aroha is a structural engineer designing a triangular steel truss for a new footbridge. One of the angles in the truss is formed by two beams meeting at an angle of 105°. To calculate the stress distribution, Aroha needs the exact value of cos(105°). Using the angle subtraction formula for cosine, where cos(A - B) = cos(A)cos(B) + sin(A)sin(B), express 105° as the difference of two standard angles (from 180°, 135°, 120°, 90°, 60°, 45°, 30°, 0°) whose trigonometric values are known. Derive the exact simplified value of cos(105°) using this approach. Answer: -(sqrt(6) - sqrt(2))/4 Solution: Express 105° as a difference of standard angles. 105° = 135° - 30°. Use the angle subtraction formula for cosine: cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
   Full step-by-step solution

   Step 1: Express 105° as a difference of standard angles. 105° = 135° - 30°. Step 2: Use the angle subtraction formula for cosine: cos(A - B) = cos(A)cos(B) + sin(A)sin(B). Here A = 135° and B = 30°. Step 3: cos(135°) = -sqrt(2)/2, sin(135°) = sqrt(2)/2. Step 4: cos(30°) = sqrt(3)/2, sin(30°) = 1/2. Step 5: Substitute: cos(105°) = cos(135° - 30°) = cos(135°)cos(30°) + sin(135°)sin(30°). Step 6: cos(105°) = (-sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2). Step 7: Multiply: cos(105°) = -sqrt(6)/4 + sqrt(2)/4. Step 8: Combine: cos(105°) = (sqrt(2) - sqrt(6))/4. Step 9: Factor the negative: cos(105°) = -(sqrt(6) - sqrt(2))/4. The answer is -(sqrt(6) - sqrt(2))/4.

4. Charlotte is a civil engineer designing a curved road that transitions from one direction to another. The transition curve is defined by an angle of 105°, which she needs to analyze using trigonometric identities. She decides to derive the exact value of sin(105°) by expressing it as the sum of two standard angles (60° and 45°) and applying the sine addition formula. Derive the exact value of sin(105°) in simplified radical form, showing all steps of the derivation using the formula sin(A + B) = sin A cos B + cos A sin B. Answer: (sqrt(6) + sqrt(2))/4 Solution: Express 105° as the sum of 60° and 45°, since 60° + 45° = 105°. Step 2: Apply the sine addition formula: sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°).
   Full step-by-step solution

   Step 1: Express 105° as the sum of 60° and 45°, since 60° + 45° = 105°. Step 2: Apply the sine addition formula: sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°). Step 3: Substitute the exact values: sin(60°) = sqrt(3)/2, cos(60°) = 1/2, sin(45°) = sqrt(2)/2, cos(45°) = sqrt(2)/2. Step 4: Compute each term: sin(60°)cos(45°) = (sqrt(3)/2)(sqrt(2)/2) = sqrt(6)/4. cos(60°)sin(45°) = (1/2)(sqrt(2)/2) = sqrt(2)/4. Step 5: Add the terms: sin(105°) = sqrt(6)/4 + sqrt(2)/4 = (sqrt(6) + sqrt(2))/4. The exact value of sin(105°) is (sqrt(6) + sqrt(2))/4.

5. Olivia is an astronomer studying the motion of a star that appears to trace a path across the sky. She models the star's angular position above the horizon at time t hours after midnight using the function θ(t) = 45° + 15t°. To determine the exact height of the star at t = 1 hour, she needs to compute sin(60°). However, she only knows the exact trigonometric values for 30°, 45°, and 90°. Using the angle subtraction formula for sine, derive sin(60°) by expressing 60° as the difference of two known angles (e.g., 90° - 30°), and then find its exact value. Answer: √3/2 Solution: Express 60° as a difference of two standard angles: 60° = 90° - 30°. Substitute: sin(60°) = sin(90° - 30°) = sin 90° cos 30° - cos 90° sin 30°.
   Full step-by-step solution

   Step 1: Express 60° as a difference of two standard angles: 60° = 90° - 30°. Step 2: Apply the sine subtraction formula: sin(A - B) = sin A cos B - cos A sin B, where A = 90° and B = 30°. Step 3: Substitute: sin(60°) = sin(90° - 30°) = sin 90° cos 30° - cos 90° sin 30°. Step 4: Use known exact values: sin 90° = 1, cos 30° = √3/2, cos 90° = 0, sin 30° = 1/2. Step 5: Substitute these values: sin(60°) = (1)(√3/2) - (0)(1/2) = √3/2 - 0 = √3/2. Step 6: Therefore, the exact value of sin(60°) is √3/2.