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Katie’s is building a barn, as shown. She wants the roof height to be between 4 and 6 feet. She must decide the angle measure to use for the pitch, or slant, of the roof when the slant height is y feet. Write an inequality Katie can use to ensure that her roof will be within the necessary height range.

To find the height of the roof, use a trig ratio like sine, cosine, or tangent. Recall that each ratio is defined as specific sides in relation to an angle.

Sine is opposite over hypotenuse.
Cosine is adjacent over hypotenuse.
Tangent is opposite over adjacent.

Each side opposite, adjacent and hypotenuse is found based on their location in the right triangle and according to the position of the angle. Since height is the value needed to be found, divide the roof in half to mark the height of the roof. This will form two right triangles. Angle x is formed by the hypotenuse and one leg of the right triangle. The leg which forms the angle is 20 since it is half the width of the barn 40. Use the tangent ratio, since the side 20 and height in respect to angle x are its adjacent and opposite side. The height must be between 4 and 6 so this will be the left and right boundary for the inequality.

4 ≤ h ≤ 6

Substitute h = 4 and h= 6 into the tangent expression.

[tex]\frac{opposite}{adjacent} \leq tan x \leq \frac{opposite}{adjacent} \\\\\frac{h}{20}\leq tan x \leq \frac{h}{20}[/tex]

[tex]\\\frac{4}{20} \leq tan x \leq \frac{6}{20}[/tex]

You can simplify each fraction by reducing using common factors.

[tex]\frac{1}{5} \leq tan x\leq \frac{3}{10}[/tex]