Addition Formulas Worksheets Grade 11

1. Using the angle addition formula for sine, find the exact value of sin(75°) by expressing it as sin(45° + 30°).
2. sin(165°)cos(15°) - cos(165°)sin(15°) = ?
3. Isabella is a marine biologist studying the path of a migrating whale pod. The pod's movement relative to a research station can be modeled by the vector sum of two displacement vectors: one 17 km at an angle of 82° north of east, and another 7 km at an angle of 37° north of east. To calculate the total displacement, Isabella needs to compute cos(82° - 37°) exactly. Using the cosine subtraction formula, prove that cos(82° - 37°) = cos(82°)cos(37°) + sin(82°)sin(37°), and then determine the exact value of cos(82° - 37°).

…and 4 more problems

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.
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)?
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.

…and 3 more problems

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.
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.
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.

…and 2 more problems