Question

can anyone help me with this A building lot in a city is shaped as a 30° -60° -90° triangle. The side opposite
the 30° angle measures 41 feet.
 
a. Find the length of the side of the lot opposite the 60° angle.

b. Find the length of the hypotenuse of the triangular lot.

c. Find the sine, cosine, and tangent of the 30° angle in the lot. Write your
answers as decimals rounded to four decimal places.

Answer 1

This is a right triangle, so we can use our standard trig definitions.

Answer 2

To find the base of the triangle (the side opposite the 60 degree angle), we can use the definition of tangent as opposite/adjacent.

\[\tan 30^\circ = 41/b\]where \(b\) is the length of the base
\[b =\frac{41}{\tan 30^\circ}\]

To find the length of the hypotenuse, we can use the Pythagorean theorem, where \[h = \sqrt{b^2 + 41^2}\]We could also use the trigonometric definition of the sin of 30 degrees or the cosine of 60 degrees.
\[\sin 30^\circ = 41/h\]\[\cos 60^\circ = 41/h\]or other combinations I won't bother enumerating. With these latter options, you'll have to solve for \(h\) in one of those equations.

Finally, you can use the basic definition of sin, cos, tan to find the answers for part c, or your knowledge of the unit circle.