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# (Originais Teachy 2023) - Question Medium of Math

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Equations and Inequalities: Trigonometric
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Question 1:

Medium

On one of the starry nights that you observed with the naked eye, you noticed that the Orion constellation was visible and decided to calculate the height of Betelgeuse, one of the brightest stars in the constellation, relative to the horizon. Knowing that the angle formed between the horizon line and the line pointing to Betelgeuse is 45° and that the distance from Earth to Betelgeuse is approximately 640 light-years, what is the height of Betelgeuse relative to the horizon? Use the tangent of the 45° angle in the resolution.
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Question 2:

Medium

Consider a building with windows in the shape of a semicircle. The construction department wants to paint the portion of the wall where the windows occupy an area not exceeding 40% of the area of the semicircle. If these windows are arranged side by side and have an angle θ (theta), such that 0º < θ < 180º, solve the trigonometric inequality that represents the described situation and determine in which interval the angle θ should be, in degrees.
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Question 3:

Hard

Observe the sequence of figures below, built with black and white balls, all of the same size. "Figure 1 is composed of 1 white ball surrounded by 8 black balls. Figure 2, 4 white balls surrounded by 12 black balls and so on." The number of black balls in the figure composed of 676 white balls is
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Question 4:

Medium

An engineer wants to calculate a specific frequency in an electrical circuit involving sines and cosines. He needs to simplify the circuit expression using the Prostaférese Formulas. Help the engineer simplify the following expression: sin(x)cos(y) - cos(x)sin(y).
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