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.

Tangent is positive in two quadrants: quadrants 1 and 3 (where sine and cosine have the same sign as each other)

Cosine is negative in two quadrants: quadrants 2 and 3 (the left quadrants, where x is negative).

The only quadrant that shares both qualities is quadrant 3.

When drawn from the origin, a line 45 degrees above (counterclockwise from) the positive x-axis lies in quadrant I. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant I composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies along the positive x-axis. Since the adjacent side and hypotenuse are both represented by positive numbers, the fraction A/H is positive. Therefore, cos 45 is positive.

When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Sine is defined as the ratio between the opposite side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. Since the opposite side and hypotenuse are both represented by positive numbers, the fraction O/H is positive. Therefore, sin 135 is positive.

When drawn from the origin, a line 135 degrees counterclockwise from the positive x-axis lies in quadrant II. Tangent is defined as the ratio between the opposite and adjacent sides of a triangle. A right triangle can be drawn in quadrant II composed of any point on that line, the origin and a point on the x-axis. The opposite side of this triangle lies above the x-axis, and is therefore represented by a positive number. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and adjacent side have opposite signs, the fraction O/A is negative. Therefore, tan 135 is negative.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant III. Cosine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and hypotenuse have opposite signs, the fraction A/H is negative. Therefore, cos 225 is negative.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant III. Sine is defined as the ratio between the adjacent side of a triangle and the hypotenuse of the triangle. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The hypotenuse of this triangle is considered a length, and is therefore positive. The opposite side of this triangle lies below the x-axis, and is therefore represented by a negative number. Since the opposite side and hypotenuse have opposite signs, the fraction O/H is negative. Therefore, sin 225 is negative.

When drawn from the origin, a line 225 degrees counterclockwise from the positive x-axis lies in quadrant II. Tangent is defined as the ratio between the opposite and adjacent sides of a triangle. A right triangle can be drawn in quadrant III composed of any point on that line, the origin and a point on the x-axis. The opposite side of this triangle lies below the x-axis, and is therefore represented by a negative number. The adjacent side of this triangle lies on the negative x-axis, and is therefore represented by a negative number. Since the opposite side and adjacent side have the same sign, the fraction O/A is positive. Therefore, tan 225 is positive.