How to Solve Trigonometry Problems with Cosine
The cosine of a non-right angle in a right triangle is the ratio of the adjacent leg length to the hypotenuse length. It is related to the trigonometric function sine by the equations cos(θ)^2 = 1 - sin(θ)^2 (degrees or radians), and cos(θ) = sin(π/2 - θ) (radians). It is also related to the tangent function by the relation sin(θ)/cos(θ) = tan(θ). These three equations are fundamental trigonometric relations.
The inverse of the cosine function is written arccos() or cos-1(). If X is the length of the adjacent side and Y is the hypotenuse, then θ = arccos(X/Y). You can use both cosine and its inverse function to solve for angles and lengths in a variety of geometric and trigonometric problems. Try the following example problems with solutions for more practice.
Example 1: Two-Part Ramp
Sara is building a ramp with two pieces of board that are 17 feet long each. The first board will be set to make an angle of 25 degrees with the ground. The second board will be set so that it makes a convex angle of 170 degrees with the first board. What is the total horizontal length of the ramp? What is the total height?
Solution: In the diagram below, the side view of the ramp system is a quadrilateral that can be subdivided into two right triangles (yellow and pink) and a rectangle (gray). The widths and heights of these right triangles are found using cosine and sine.
The pink right triangle has an angle of 25 degrees and a hypotenuse of 17, therefore its width is 17*cos(25) feet and its height is 17*sin(25) feet.
To find the angle of the yellow triangle, we compute 170 - (90-25) - 90, which gives us 15 degrees. Therefore, the yellow triangle has a width of 17*cos(15) feet and a height of 17*sin(15) feet.
The total horizontal length of the ramp is 17*cos(25) + 17*cos(15) ≈ 31.83 feet, and the total height is 17*sin(25) + 17*sin(15) ≈ 11.58 feet.
Example 2: Dihedral Angle of Square Pyramid
A square pyramid has a slanted side length of 7 meters and a base length of 5 meters. What is the dihedral angle between the square base and the triangular sides, i..e., the angle between the two planes? This is also called the slant angle of the pyramid's triangular sides.
Solution: We can solve this problem by first cutting the pyramid in half going parallel to one side of the square base. This gives us two copies of a rectangular pyramid with three slanted sides and one vertical side, outlined in red. The dimensions of the bases are 5 meters by 2.5 meters.
The slanted side length of the red triangle is found using the Pythagorean Theorem on a right triangle with a leg of 2.5 and a hypotenuse of 7. This gives sqrt(7^2 - 2.5^2) = sqrt(42.75).
Cutting the red triangle in half along its altitude gives another right triangle with a leg length of 2.5 and a hypotenuse of sqrt(42.75). Examining this right triangle with an angle of θ at the base, we can see that the angle θ satisfies the relation
cos(θ) = adjacent/hypotenuse = 2.5/sqrt(42.75)
This works out to
θ = arccos(2.5/sqrt(42.75))
θ ≈ 67.52 degrees
The angle θ is the dihedral angle between the base of the pyramid and its sides. Thus, the sides slant at an angle of 67.52 degrees from the base.
Example 3: Dog on a Rope
Sandra's back yard is a rectangle that measures 20 feet along the back fence. She needs to tie up her dog along the back fence with 14 feet of rope while she eats dinner with friends on the back porch. She realizes that if she stakes the end of the rope at either corner of the fence, her dog will have a quarter circle of space to roam freely, with an area of (π/4)(14^2) = 153.94 square feet (shown in blue and pink). But if she stakes the end of the rope at the midpoint of the back fence, the dog will have much more space to roam (shown in green). How much space will the dog have if she stakes the end of the rope at the center of the back fence?
Solution: The green region can be broken up into simpler shapes whose areas are easier to compute. If you look at the diagram below, you will see that it can be partitioned into two equal right triangles and a circular sector.
The combined area of the right triangles is 10*sqrt(96) ≈ 97.98 square feet.
To find the area of the circular sector, we need to find the angle θ and plug it into the expression
We can find θ by finding the angles φ of the isosceles triangle. Cutting the isosceles triangle in half produces two right triangles, each with a leg length of 10 feet and a hypotenuse of 14 feet. This means cos(φ) = 10/14, and φ = arccos(10/14) ≈ 44.42 degrees. Now using the triangular angle sum equation
θ + 2φ = 180
we obtain θ ≈ 91.17 degrees. Now we can compute the area of the circular sector:
(91.17/360)*pi*14^2 ≈ 155.94 square feet.
The total area for the dog to roam freely is 97.98 + 155.94 = 253.92 square feet. Compared to the roaming area provided by staking the end of the rope in the corner of the yard, this is an increase of about 65%.