Question bank: Trigonometric Lines of 75º, 15º, 22.5º

During a Physics class, a group of students decides to analyze the movement of a Ferris wheel to determine the maximum height that one of its cabins reaches. Considering that the diameter of the Ferris wheel is 50 meters and that, in the initial position, the cabin is 25 meters above the ground, they want to calculate the maximum height reached by this cabin during a complete cycle of movement. Knowing that the height of the cabin relative to the ground can be modeled by a sinusoidal function, the students choose a coordinate system in which the center of the Ferris wheel corresponds to the origin and the positive vertical axis points upwards. If the movement is described by the function h(t) = A * sin(Bt + C) + D, where h(t) is the height of the cabin at time t, A is the amplitude of the movement, B influences the period of the movement, C represents the initial phase of the movement, and D is the vertical equilibrium position, determine the function that describes the height of the cabin relative to the ground and calculate the maximum height reached by it. Consider that the complete up and down movement of the Ferris wheel is done in π/2 hours.

During an interdisciplinary project of Architecture and Mathematics, a group of high school students is working on the design of a stained glass window for a modernist building. The envisioned stained glass window has a radially symmetric design, and to calculate the precision of the glass cuts, it is necessary to determine the exact dimensions of each sector of the stained glass window. One of the sectors is an angular circular segment with a central angle of 75°, extending from the center of the circle to its circumference. To ensure the visual harmony of the stained glass window, the students need to calculate the exact measure of the radius of this circular sector. Considering the knowledge of trigonometric functions of the given angles and the relationship between the arc length and the radius to determine the dimensions of the circular sector, what is the exact value of the radius that the students should use to draw this sector of the stained glass window?

In an engineering project, an architect needs to calculate the exact dimensions of a set of pillars that will support a triangular structure. To ensure the stability of the construction, the architect decides that the angle formed between the ground and the pillars should be 15º. If the pillar directly intersects the vertex of the triangle's base, which measures 8 meters, and the responsible engineer wants to know the exact height the pillar will reach before being cut for the final finish. What is the height that the pillar will reach in relation to the ground, and how can the trigonometric lines of 15º be applied to solve this situation?

During a Physics experiment, a student observes that, in order to make precise measurements, it is necessary to calculate trigonometric values for 75º, 15º, and 22.5º. Given the context, (a) demonstrate how to calculate the trigonometric values for 75º (sine, cosine, and tangent) from the trigonometric values of 45º and 30º. (b) Then, do the same for the trigonometric values of 15º from the trigonometric values of 45º and 60º. (c) Finally, calculate the trigonometric values of 22.5º using the trigonometric values of 45º and the trigonometric angle sum identity.

Consider a lighthouse located on an island that emits a beam of light that varies its direction over time following a trigonometric pattern. If at a given moment the lighthouse is directed towards a point P in the water, and the light line forms an angle of 75 degrees with the surface of the water, determine the point Q that will receive the light reflected on the surface of the water if it forms an angle of 15 degrees with the incident line. Then, determine point R on the light line that starts from the lighthouse and forms an angle of 22.5 degrees with the light line incident on point Q. To do this, use sines and cosines of the given angles, assuming that light travels in a straight line and that there is no refraction. Explain the reasoning step by step, highlighting the use of trigonometric relationships for each step of the calculation.