Trigonometric Identities Worksheets Grade 12

1. Verify: (csc²θ - cot²θ) / (sec²θ - tan²θ) = 1.
2. Liam is designing a suspension bridge where the main cable follows the curve y = 15sin(x/10) + 20, where y is the height in meters above the water and x is the horizontal distance in meters from the left tower. Engineers need to know the steepest slope of the cable to design proper connections. What is the maximum slope of the cable?
3. A circular oil spill is expanding such that its radius increases at a constant rate of 0.5 meters per second. At a particular moment, the radius is 12 meters. Using calculus, determine the rate at which the area of the oil spill is increasing at that moment. (Use π in your calculation)

…and 2 more problems

1. A right circular cone has a height of 15 cm and a base radius of 8 cm. A plane parallel to the base cuts the cone, creating a smaller similar cone at the top with a height of 5 cm from the vertex. What is the volume of the frustum (the remaining portion between the two parallel planes)? Use π in your calculation.
2. An engineer is designing a roller coaster with a track section that follows the path y = 3sin(x) - 4cos(x), where y is the height in meters and x is the horizontal distance in meters. To ensure safety, she needs to verify that this path can be rewritten in the form y = Rsin(x - α) where R > 0. Find the exact values of R and α that satisfy this identity.
3. Verify: (sin⁶θ - cos⁶θ) / (sin²θ - cos²θ) = 1 + sin²θ cos²θ.

…and 4 more problems

1. A civil engineer is designing a suspension bridge where the main cable forms a parabolic curve described by the function f(x) = ax² + bx + c. The cable passes through points (-100, 50), (0, 10), and (100, 50) relative to the bridge's center. The engineer needs to verify that the cable's shape satisfies the trigonometric identity sin²θ + cos²θ = 1 at the point where the cable makes a 30° angle with the horizontal. Find the value of sin²θ + cos²θ at this point to confirm the identity holds.
2. An engineer is designing a roller coaster with a drop that follows the path y = 25sin(x/10) + 15cos(x/10), where y is the height in meters and x is the horizontal distance in meters from the start of the drop. To ensure rider safety, she needs to verify that this path can be rewritten in the form y = Rsin(x/10 + α), where R > 0 and α is a phase shift. Find the amplitude R of this transformed function, which represents the maximum deviation from the average height.
3. Verify: (sec²θ - 1) / (tan²θ + 1) = sin²θ.

…and 3 more problems