Last Laughs....

Example #1

Example:
You want to build stairs to the fourth story in an apartment building. The distance from the floor and the top of the stairs on each floor is 13 feet. You decide you want to makes the stairs at a 45 degree angle. You need to know how long to make the railing for all four stories. Assume that you need railings on only one side of the stairs, and that you don’t need railing on the “ends” of the stairs.  The amount of railing is equal to the “bottom” of the stairs.

Step 1: Since we know one angle measure, and it’s opposite leg, we can use the trigonometric function of sine (opposite/hypotenuse) to find the “hypotenuse”:

Sin(45) = 13

                                                                        H

Step 2: Multiply by “h” on both sides:

(h)(sin45) = 13

Step 3:  Divide both sides by sin 45

h= 13

       sin45

Step 4: Multiply 13/sin45 by 4 (for the four flights of stairs):

 (4)13

(1)sin45

Step 5: Simplify :

      52___

                                                                Sin45       equals approximately  73.54

Step 6:  You can’t have 73.54 feet of railing, so you need 74 feet of railing for all four stories.

Last Laughs...

Application:Physics

Application: Physics

Trigonometry can be found in many areas of science, including physics. Recall that the Unit Circle is helpful in finding common trigonometric identities, and therefore, the Unit Circle can sometimes be used in physics. In this example, we see a father pulling his children with a rope attached to a sled. If we can know that there is 50 N **of force on the rope, we can break it down even further and specify the vertical force and horizontal force working on the rope.

** A newton is the SI (International System of Units) used for magnitude, or strength, of a force

Figure 6: Using trigonometry to resolve forces

Looking at Figure 6b and applying the trigonometry from earlier in this section, we can say

                            AND                              

Rearranging these two equations gives

      AND                        

The horizontal component of the force is approximately 43.30 N.

The vertical component of the force is 25 N.

Lets Apply: Architecture!

The Unit Circle has a lot of real life applications. Sine, Cosine and Tangent are used frequently in construction.


     Surveyors need to know the height of a building in order to build buildings nearby with the same height measure. (This way, the buildings don't look awkward when they are built next to each other!)
There is a very simple way to figure this out:
First, the surveyor needs to stand a certain distance from the side of the building they want to find the height of. The surveyor should then , measure that distance in the units they want to find the height of the building in.
From the point where the surveyor is standing, draw an imaginary line upward, and using surveying tools, you can figure out that measure (called measure of elevation).
Then, use tangent and the measure of elevation, or theta. to find the height of the building:
     Tangent(theta)=A/B

Multiply the B out on both sides:
    Btangent(theta)=A


     To complete a roof, constructors need to know how long the roof is to know how long the boards that will cover the roof will be. (That way, your house will have even boards that cover the roof!)
There is a very simple way to figure this out:
First, the constructor needs to know the angle at the top of the roof...divided by two. We will call this angle theta. Also, the constructor needs to know either half the length of the base of the roof OR the height of the roof.
Then, use sine or cosine (depending on what given information you have to solve the length of the top of the roof.

     If you know the height of the roof:

Cos(theta)=h/length of roof  and multiply the length of roof on both sides:

length of roof(cos)theta=h  remember we're solving for the length of roof...divide both sides by cos(theta):

length of roof=h/cos(theta)

If you know the half the base of the roof (b):

sin(theta)=b/length of roof  and multiply the length of roof on both sides:

length of roof(sin)theta=b  remember we're solving for the lenght of roof...divide both sides by sin(theta):

length of roof=b/sin(theta)

Remember that there is a way to find sine, cosine and tangent measures of 30-60-90 and 45-45-90 triangles easily! (This "trick" is how we plot points on the unit circle!) You can apply this to get exact measurements when constructing, to get the most accurate results possible.)



Who discovered the unit circle?

     There is actually no one person, that took credit for discovering the unit circle, but there are aspects of the unit circle that other people previously discovered.
     The unit circle lies on the Cartesian coordinate system , which Rene Descartes discovered in 1637. It is a way to find points in a plane using two points called x-coordinate and y-coordinate.
The Cartesian Coordinate System is in the Euclidean plane, which was discovered by a Greek Mathemetician named Euclid.
Radians are used to represent angle measures in the unit circle. Roger Cotes, an English Mathematician created this concept of using Radian measures to represent angle measures in 1714.

Haha...