8-1

In this section, we learn how to find the angle of inclination. M=tan(a); m stands for the slope of a line and a stands for alpha which is also known as the angle of inclination. To find the angle of inclination of a conic, you use tan2a=B/A-C.

Ex. 1 5cos(theta)+10=7

1. First, you subtract 10 which lives you with 5cos(theta)=-3.
2. Now, you divide 5 by on both sides which leaves you with cos(theta)=-3/5.
3. Now you you solve for theta and take the inverse of cos. Theta=cos-1(-3/5).
4. Now you have to find out what cos-1(-3/5) is and what quadrant it is in. It equals to 53.130 which is in quadrant one. *Remember that you don't put the negative in front of 3/5 when you plug it in your calculator*.
5. You have to find what quadrants cos is negative. Cos is negative in quadrants two and three. Now, in order to move from quadrant one to quadrant two, you have to make 53.130 negative and then add 180. You end up with 126.87 in quadrant two. Now you move from quadrant one to quadrant three. In order to do that, you just add 180 which gives you 233.13.
6. Since the numbers in quadrant two and three are decimals, you have to convert them into degrees, minutes, and seconds. To convert 126.87 to degrees, minutes, and seconds, take .87 and multiply it by 60. That gives you 52.2. Now you take .2 and multiply it by 60 which gives you 12. Now you do the same thing with 233.13. .13x60=7.8 and .8x60=48.
7. Final answer: Theta=126 degrees 52 minutes 12 seconds and 233 degrees 7 minutes 48 seconds.

Ex. 2 Find the angle of inclination of 4x+8y=16.

1. First, you have to know that you are using the formula: M=tan(a).
2. Since the equation is a line, you have to find the slope. M=-A/B=-4/8=-1/2.
3. Now you can plug that into the equation: tan(a)=-1/2.
4. Now you solve for alpha and take the inverse of tan: a=tan-1(-1/2).
5. You find out what tan-1(-1/2) equals to: 26.565. Now you find out what quadrants tan is negative in: two and four.
6. Find quadrant two: -26.565+180=153.435. Find quadrant four: -26.565+360=333.435.
7. Turn both of those answers into degrees, minutes, and seconds: .435x60=26.1 .1x60=6 (I only did it once since both of the decimals end with .435).
8. Final answer: A = 153 degrees 26 minutes 6 seconds and 333 degrees 26 minutes and 6 seconds.

Ex. 3 Identify the graph of the equation x^2-5xy+8y^2=1 and find the angle of inclination.

1. First, you have to find out what kind of conic this equation is: B^2-4(a)(c)=(-5)^2-4(1)(8)=168. Since it is a positive number, it is a hyperbola.
2. Plug the numbers from the equation above ^^^^ into tan2a=B/A-C: tan2a=-5/1-8=-5/-7=5/7.
   
3. Now you solve for alpha and take the inverse of tan: 2a=tan-1(5/7).
4. Find out what tan equals to and what quadrant tan is positive in: tan-1(5/7)=35.538. Tan is positive in quadrant one and three.
5. Find out what quadrant three is: 35.538+180=215.538.
6. 2a=35.538;215.538. Divide by two on each side: a=17.769;107.769. Put in degrees, minutes, and seconds: .769x60=46.14 .14x60=8.4
7. Final answer: a=17 degrees 46 minutes 8 seconds and 107 degrees 46 minutes 8 seconds.

-Amber