Basics and Applications of Trigonometry

As shown in the following figure, the sizes of the angles when looking up at a satellite floating in the sky simultaneously from two observatories and located away are and , respectively. Find the distance from the observation station to the satellite.

In the right triangle shown below, it holds and . If and are the vertices on the extentions of and respectively, find the length of that satisfies and .

As shown in the following figure, the radius of the base is and the size of angle between the generatrix and the line of height is in a cone. Find the value of when the volume of the cone is .

When in the right triangle with , Choose the one that is correct.

As shown in the following figure, is the midpoint of in the tetrahedron whose edge length is , and the point is the foot of the perpendicular line from onto the base. Find the value of when .

A is the foot of the perpendicular from one vertex to of a cube with an edge length of , as shown in the following figure. When , find the value of .

For the right triangle shown below, choose the one that is correct.

Find the length of in the right triangle with as shown in the figure below.

As shown in the following figure, the distance between a boat sailing on the sea and the lighthouse is , and the size of the angle when looking up the top of the lighthouse from the boat is . Find the height of the lighthouse. (note, , , and the size of the boat should be ignored.)

In the figure below, find the area of when , , and .

In the figure below, is the diameter of the semi-circle . Find the area of shaded region when and

Find the area of when , , and the point is the center of mass of as shown in the figure below.