Section 5.2A. The Six Trigonometric Functions.

Given a right triangle, there are six ways to express side length ratios. The words adjacent and opposite are descriptive words that correspond to the given angle θ. The angle θ is the angle measure between the adjacent side and the hypotenuse. The opposite side is the side that does not touch θ.

Let us shorten adjacent side as ADJ, opposite side as OPP, and the hypotenuse as HYP. The six trig ratios are: OPP ADJ OPP ADJ HYP HYP sin(θ) = cos(θ) = tan(θ) = cot(θ) = sec(θ) = csc(θ) = HYP HYP ADJ OPP ADJ OPP From this chart, the following are evident: 1. sin(θ) and csc(θ) are are reciprocals. So are the pairs cos(θ) & sec(θ) and tan(θ) & cot(θ). 2. Because the hypotenuse is the largest side, sin(θ) and cos(θ) cannot be larger than 1. This is because the numerator will never be larger than the denominator. 3. Similarly, sec(θ) and csc(θ) will never be smaller than one. This is because the numerator will never be smaller than the denominator. Exercises: 1. You are given a right triangle whose legs have lengths a and b. Compute trig(θ) in terms of a and b. 2. A right triangle has hypotenuse 8 and an acute angle of 20◦. Determine the area and the perimeter of the triangle. 3. The base of a right circular cone has diameter 16 inches. The angle between the radius and the slanted edge is 68◦. Determine the height of the cone and the volume of the cone. 4. A radio transmission tower is 36 feet tall and makes a right angle with the ground. A guy wire is to be attached at the top of the tower and makes an angle of 48◦ with the ground. Determine the length of the guy wire. 5. In the diagram below, you are given that BD = 10 and BE = 4. Compute the outer perimeter of the shaded region in terms of θ. 6. In the diagram below, you are given that BD = 3 and CE = 11. Compute the outer perimeter of the shaded region in terms of θ. Special Triangles. There are two special triangles that you will need to be familiar with. They are the 45◦ − 45◦ − 90◦ triangle and the 30◦ − 60◦ − 90◦ triangles.

Examples: Use the triangles above to compute the exact values for the following:

sin(30◦) = sin(45◦) = sin(60◦) =

cos(30◦) = cos(45◦) = cos(60◦) =

tan(30◦) = tan(45◦) = tan(60◦) =

cot(30◦) = cot(45◦) = cot(60◦) =

sec(30◦) = sec(45◦) = sec(60◦) =

csc(30◦) = csc(45◦) = csc(60◦) =