Trigonometric Equations Worksheets Grade 11

1. Ava is an astronomer studying the light curve of a variable star. The brightness of the star over time is modeled by the equation B(t) = 6 sin^2(t) - 11 sin(t) + 6, where t is the phase angle in radians measured from 0 to 2π. Ava needs to find all phase angles t in the interval [0, 2π) where the brightness drops to zero. Solve the trigonometric equation to help Ava identify these critical phase angles.
2. A radio tower is anchored by two guy wires on opposite sides. The first wire makes a 60° angle with the ground and is attached 50 meters from the tower's base. The second wire makes a 45° angle with the ground. If both wires reach the same point at the top of the tower, what is the height of the tower?
3. Solve: 2cos²(x) - 5cos(x) - 3 = 0 for x ∈ [0, 2π]

…and 4 more problems

1. Aroha is an aerospace engineer modeling the vibration of a satellite's solar panel. The panel's angular displacement θ (in radians, from 0 to 2π) is governed by the equation 9sin²θ - 12sinθ + 4 = 0. Find all possible angular displacements θ in the interval [0, 2π) that satisfy this equation.
2. An engineer is designing a suspension bridge where the main cable forms a parabolic shape approximated by the function y = 25 cos(πx/100) + 15, where y is the height in meters above the road deck and x is the horizontal distance in meters from the center of the bridge. The engineer needs to install vertical support cables at the points where the main cable is exactly 30 meters above the road deck. Find all values of x between -50 and 50 meters where this occurs.
3. Matiu is an acoustic engineer designing a noise-canceling system for a factory floor. Two machines produce sound waves that interfere, and the combined sound pressure at a sensor is modeled by the equation 2sin²θ - 3sinθ + 1 = 0, where θ is the phase difference in radians between the waves, measured from 0 to 2π. Find all possible phase differences θ in the interval [0, 2π) that result in zero sound pressure at the sensor.

…and 3 more problems

1. Matiu is a marine biologist studying the rhythmic flashing patterns of a rare species of jellyfish. The light intensity emitted by a jellyfish is modeled by the equation 4 sin²θ - 2 sinθ - 2 = 0, where θ is the phase angle of the jellyfish's internal biological clock, measured in radians from 0 to 2π. Find all possible phase angles θ in the interval [0, 2π) that cause the light intensity to drop to zero.
2. A right triangle is inscribed in a unit circle such that the hypotenuse is the diameter. If one acute angle measures θ, and the point where the terminal side intersects the circle has coordinates (x, y), express the area of the triangle in terms of trigonometric functions of θ.
3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is circumscribed around this triangle such that all three vertices lie on the circle. What is the exact length of the radius of this circumscribed circle?

…and 3 more problems