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

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Question 1:

Medium

An architect is designing a roof and is using angles of 15º, 22.5º, and 75º in his project. Determine the exact value of the sine of the 75º angle.

Trigonometric Lines of 75º, 15º, 22.5º

Question 2:

Easy

In an engineering project, a beam of light needs to be directed so that it strikes a reflective surface at an angle of 75º in relation to the normal of that surface, in order to avoid dazzling operators who will be handling parts in the same environment. The light initially strikes perpendicular to the ceiling, and a mirror placed at 15º in relation to this light source will reflect it in the desired direction. Additionally, it is necessary to install a prism with an angle of 22.5º so that the light coming out of the mirror reaches the reflective surface with the correct inclination. Considering only the action of the mirror and the prism on the direction of the light beam, calculate the angle at which the light exits the mirror and the additional rotation that the light beam undergoes when passing through the prism.

Trigonometric Lines of 75º, 15º, 22.5º

Multidisciplinary Question

Question 3:

Easy

The value of cos 105º is

Trigonometric Lines of 75º, 15º, 22.5º

Basic Calculus

Question 4:

Medium

In a real situation, a graphic designer wants to tilt an object at an angle of 15º relative to the horizontal. He designed a trigonometric line to assist him in this process, but he faced the challenge of calculating the sine and cosine of this angle. Using your knowledge in the area and considering the method of notable angles, find the value of sin(15º) and cos(15º), respectively, in terms of radical fractions.

Trigonometric Lines of 75º, 15º, 22.5º

Question 5:

Hard

During the cleaning of the cooling system of a factory, a technician must adjust a light reflector fixed on the ceiling of the room. The reflector has the shape of a right circular cone with the vertex pointing upwards and an opening angle of 75º. The technician wants to calculate the height and the radius of the base of the cone to determine the ideal dimensions of the mirrors that will be installed on the inner surface of the cone, so that the reflected light follows a beam parallel to the floor. Consider that the angle of incidence of light on the inner surface of the cone should be 15º with the normal to the surface at the point of incidence to ensure total reflection of light. Based on this configuration, determine the height and the radius of the base of the cone, and additionally, identify the value of the sine and cosine of 22.5º, an angle that arises in the resolution of the problem.

Trigonometric Lines of 75º, 15º, 22.5º

Multidisciplinary Question

IARA TIP

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