To solve this problem, we need to create a diagram, but in order to create that diagram, we need to understand the vocabulary that is being used in this question. The following diagram clarifies the difference between an angle of depression (an angle that looks downward; relevant to our problem) and the angle of elevation (an angle that looks upward; relevant to other problems, but not this specific one.) Imagine that the top of the blue altitude line is the top of the lighthouse, the green line labelled GroundHorizon is sea level, and point B is where the boat is.

Merging together the given info and this diagram, we know that the angle of depression is 19^o and and the altitude (blue line) is 105 meters. While the blue line is drawn on the left hand side in the diagram, we can assume is it is the same as the right hand side. Next, we need to think of the trig function that relates the given angle, the given side, and the side we want to solve for. The altitude or blue line is opposite the known angle, and we want to find the distance between the boat (point B) and the top of the lighthouse. That means that we want to determine the length of the hypotenuse, or red line labelled SlantRange. The sine function relates opposite and hypotenuse, so we'll use that here. We get:

 (where d is the distance between the top of the lighthouse and the boat)

 (using a calculator in degree mode and rounding to two digits, we get that )

 meters

As with other trig problems, begin with a sketch of a diagram of the given and sought after information.

Angelina and her car start at the bottom left of the diagram. The road she is driving on is the hypotenuse of our triangle, and the angle of the road relative to flat ground is 22^o. Because we want to find the change in height (also called elevation), we want to determine the difference between her ending and starting heights, which is labelled x in the diagram. Next, consider which trig function relates together an angle and the sides opposite and hypotenuse relative to it; the correct one is sine. Then, set up:

 (using a calculator in degree mode and rounding to two decimals we get that )

Therefore the change in height between Angelina's starting and ending points is 1480 meters. 

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding :

Remember that this is not the full height of the larger building. To find that, we need to add  feet. Therefore, the taller building is 95.5 feet tall.

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding :

Remember that this is not the full height of the larger building. To find that, we need to add  feet. Therefore, the taller building is 104.6 feet tall.

To solve this problem, let's start by drawing a diagram of the two buildings, the distance in between them, and the angle between the tops of the two buildings. Then, label in the given lengths and angle. 

We are being asked to find the height of the taller building, but this diagram does not provide a triangle that has as one of its sides the entire height of the larger (rightmost and blue) building. However, we can instead find the distance , and then add that to the 40 foot height of the shorter building to find the entire height of the taller building. Start by finding :

Remember that this is not the full height of the larger building. To find that, we need to add  feet. Therefore, the taller building is 635.3 feet tall.

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground.  We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high.  We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle.  Our base of the triangle is 3 feet and the leg is 10 feet.

And so we need a support of 10.44 feet long.

We begin by drawing a picture. Let  stand for where the plane reaches a maximum altitude of 31,000 ft. We can also assume that if we draw a line straight down from the plane that it will be perpendicular to the ground. This forms a right triangle.

Now we will use our knowledge of right triangles.  We know the trigonometric identity, .  We can plug 55 in for our  and 31,000 in for the opposite side. Solving for the hypotenuse is solving for the distance from the point of takeoff to the plane when it reaches maximum altitude.

And so the distance from point  to the plane at maximum altitude is 37,844 ft.

We begin by drawing a picture.  Let point be where your eyes are looking on the billboard and point be where you are standing on the ground.

We must solve for the height of the billboard.  We have information of the adjacent side of the right triangle formed and we wish to know the height, which will be the opposite side from you.  Using trigonometric identities, we will use .  We can plug in 500 ft for our adjacent side and 67 degrees for our .

And the height of the section of the billboard you are looking at is 1,178 ft above the ground.

The important thing to note for this problem is that we are working with similar triangles.  Your shadow forms a right triangle with its three points being your feet, the top of your head, and the top of your shadow’s head.  We have enough information to find the angle of your head’s elevation to your shadow on the ground.  Just like your shadow forms a right triangle to your body, a similar right triangle will also be formed to the sun’s altitude.  We know that this triangle is similar because it shares the angle opposite your body and it shares a right angle, so it’s third angle must also be equal.  Since it shares the angle opposite your body, we must only solve for the angle of your head’s elevation and it will be the same as the sun’s elevation.

We do this by using .  We can plug in 60cm for the opposite since that is your height, and 66cm for the adjacent since that is the length of the shadow.  We are solving for .

And so the sun is at an elevation of 48 degrees.

Since the pictured triangle is a right triangle, the unlabeled angle at the lower left is a right angle measuring 90 degrees. Since interior angles in a triangle sum to 180 degrees, the unlabeled angle at the upper left can be calculated by 180 - 60 - 90 = 30. The pictured triangle is therefore a 30-60-90 triangle. In a 30-60-90 triangle, the ratio between the hypotenuse and the shortest side length is 2:1. Therefore, C = 2A.