91Ó°ÊÓ

Angle of Depression (Opera). The balcony seats at the opera house have an angle of depression of \(55^{\circ}\) to center stage. If the horizontal (ground) distance to the center of the stage is 50 feet, how far are the patrons in the balcony to the singer at center stage?

Bearing (Navigation). If a plane takes off bearing \(\mathrm{N} 35^{\circ} \mathrm{E}\) and flies 3 miles and then makes a left turn \(\left(90^{\circ}\right)\) and flies 8 miles farther, what bearing will the traffic controller use to locate the plane?

Bearing (Navigation). If a plane takes off bearing \(\mathrm{N} 48^{\circ} \mathrm{W}\) and flies 6 miles and then makes a right turn \(\left(90^{\circ}\right)\) and flies 17 miles farther, what bearing will the traffic controller use to locate the plane?

As \(\theta\) increases from \(0^{\circ}\) to \(90^{\circ}\), how does \(\csc \theta=\frac{1}{\sin \theta}\) change?

Obstacle Course. As part of an obstacle course, participants are required to ascend to the top of a ladder placed against a building and then use a rope to climb the rest of the way to the roof. The distance traveled can be calculated using the formula \(d=15 \sin \theta+4 \sqrt{3}\), where \(\theta\) is the angle the ladder makes with the ground and \(d\) is the distance traveled, measured in feet. Find the exact distance traveled by the participants if \(\theta=60^{\circ}\).

Hot-Air Balloon. A hot-air balloon is tethered by ropes on two sides that form a \(60^{\circ}\) angle with the ground. If the height of the balloon can be determined by multiplying the length of one tether by \(\sin 60^{\circ}\), find the exact height of the balloon when 100 -foot ropes are used.

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the length of the side opposite the \(60^{\circ}\) angle is twice the length of the side opposite the \(30^{\circ}\) angle.

The two acute angles in a right triangle must be complementary angles.

Staircase. The height, measured in feet, of a certain staircase is given by the formula \(h=15 \tan \theta\), where \(\theta\) is the pitch of the staircase. What is the height of a staircase with a pitch of \(39^{\circ} 28^{\prime} 37^{\prime \prime}\) ?

\(\sin 0^{\circ} \quad\)