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In this lesson, we will be introducing the basic trigonometric functions and discussing the relationship between the angles and sides
of a right triangle.
Often times in the real world, we encounter geometric shapes we can visualize using a triangle.
Particularly in this case, we can use a right triangle.
The reason it is a right triangle is because the bottom angle is a right angle, meaning it is 90 degrees.
We often refer to the other angles using the Greek letter theta.
How is the measure of this angle related to the various sides?
The ”Hypotenuse” is the side opposite the right angle. It is also the longest side of the triangle.
We also have the “Opposite” side, which is called the opposite side because it is the side opposite our angle.
I could have called the top angle theta, which means the opposite side would have been the bottom side.
The opposite side is relative to which angle you are looking at.
The third side is called the “Adjacent” side, again relative to the angle. It is adjacent to the angle we are looking at.
So how are these 3 sides related to the measure of the angle theta? We have 3 Trig functions that give different relationships.
The first function is Sine.
We have the relationship that the sine of the angle theta is equal to the length of the opposite side divided by the length of the hypotenuse.
The second function is Cosine. The cosine of the angle theta is equal to the length of the adjacent side divided by the length of the
hypotenuse.
The third function is Tangent. The tangent of the angle theta is equal to the ratio of the opposite side divided by the adjacent side.
How can we use these functions to solve problems? Let’s look at some examples.
Suppose an escalator is designed with a 38 degree incline angle. If the length of the escalator is 18 feet, what is the vertical height at
the top of the escalator? Looking through this, we know that the incline angle is 38 degrees, which is theta.
We also know that the length of the escalator is 18 feet, which is the hypotenuse.
We are trying to find the vertical height, which is the right side or opposite side, or x.
In order to solve this problem, we need to think about which Trig function gives a relationship between the opposite side and the hypotenuse.
We had the sine of theta is equal to the opposite side over the hypotenuse.
This gives us sin(38) = x / 18. To get x by itself, multiply both sides by 18.
So x = 18 * sin(38).
Use a calculator to compute this.
First, we need to check the MODE.
We need to check if it is set to RADIAN or DEGREE. Since the problem is dealing with degrees, set it to DEGREE mode.
To compute this, we have 18 * sin(38) which is 11.08.
This means that the vertical height at the top of the escalator is 11.08 feet.
Let’s look at another example.
Suppose an escalator is designed with a 38 degree incline angle, which is theta.
If the length of the escalator is 18 feet, what is the horizontal length of the escalator?
Again the escalator, or the hypotenuse, is 18 feet.
The horizontal length is the adjacent side, which is x.
What Trig function did we have relating the adjacent side and the hypotenuse?
Cosine, so the cosine of the angle theta was equal to the adjacent side divided by the hypotenuse.
So that gives cos(38) = x / 18. Multiply both sides by 18 to cancel, which gives 18 * cos(38) = x.
This time, 18 * cos(38) is 14.18 feet, which is the measure of the bottom side.
One more example.
Suppose an escalator is designed with a 42 degree incline angle. If the horizontal length of the escalator is 20 feet, what is the height at the
top? This time, theta is 42 degrees.
The horizontal length is 20 feet, which is the bottom length, so the adjacent side is 20 feet.
The height is the opposite side, or x.
What Trig function gives a relationship between the opposite side and adjacent side?
Tangent, so the tangent of the angle theta is equal to the opposite side divided by the adjacent side.
That gives tan(42) = x / 20. Multiply both sides by 20 to get x by itself.
This gives x = 20 * tan(42).
Then 20 * tan(42) is 18.01.
So the length of the opposite side is 18.01 feet.