Law of Sines and Cosines #15

To use the law of sines, a triangle must either have 2 sides and an opposite angle or 2 angles and an opposite side. The equations to use are sinA/a = sinB/b = sinC/c. When approached with the situation of sin being attached to a letter, take the inverse of both sides. A sin and its number are not allowed to be separated because they are one pair and can only be apart it the inverse is taken. The law of cosines is only used once, then use the law of sines. To use this law, a triangle must be given 2 sides and an included x or 3 sides. The formulas for the law of cosines are a^2=b^2+c^2-2bccosA, b^2=a^2+c^2-2accosA and c^2=a^2+b^2-2abcosC. 

Chapter 4 Summary #14

Chapter 4 was a difficult one. First, we learned about angles and their measurements. Standard position is when the initial side is in the positive x-axis. Coterminous angles are angles that have the same terminal side. For complementary angles, add 90 degrees or pi/2. For supplementary angles, add 180 degrees or pi. The arc length formula us s=rtheta. Theta must be in radians and there are no exceptions! Then, we learned about special triangles: 45. 45, 90 degrees and 30, 60, 90 degrees. Then, we went into verifying trig identities. Then, the sum and difference, double, and half angle equations, which I found to be the most difficult. Inverse trig functions were relatively simpler for me. I learned to always unrationalize the arc length. 

Mr. Unit Circle #13

The unit circle is extremely helpful. It provides the cosine and sine values of the important degrees. It also gives the degrees in radians. The unit circle is a circle with a radius of 1. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. There are also trig functions on the unit circle.

Trig Equations #12

There are many different trig equations. An example of a sum and difference equation would be cos105 degrees. You must add two degrees together to make 105. Use the equation cos(x+y)=cosxcosy-sinxsiny. Cos(60+45) equals to 105 degrees, so now, you must find the sine and cosine of 60 and 45 degrees. Then, just plug them into the equation. An example of a double angle would be sinx=-1/4, solve for cos2x and sin2x. First, you would use the identity cos2x=cos^2x - sin^2x. In order to find cosx, use the identity sin^2x + sin^2x = 1. Since sin is already given, plug it in and solve for cos. 

Verifying Trig Identities #11

The first suggestion in verifying a trig identity is to simplify the more complicated side. Then, find common denominators. After that, change all the trig functions in terms of sine and cosine. Finally, use an identity! The goal in verifying trig identities is to make it equal to the other side of the equation. If you do not get it the first time, do not be discouraged and try it again. Using identities is the most important suggestion. 

Tangent #10

Tangent of theta is equal to the y value (cosine) / the x value (sine). A way to find tangent is to remember TOA. The TOA represents that tangent = opposite/adjacent. The tangent is always positive in the first and third quadrants. Tangent graphs are undefined at 90 degrees and 270 degrees and the asymptotes are at pi/2 + kpi. The periods are shorter that the sine and cosine graphs. The x intercepts equal npi. The period is pi/B. 

Sine and Cosine #9

Sine of theta is equal to the y value of a triangle. Cosine of theta is equal to the x value of a triangle. A way to find sine and cosine is to remember SOH CAH. The SOH represents sine = opposite/the adjacent side of the theta. The CAH represents cosine = adjacent/the hypotenuse of the triangle. Sine is always positive in the first and second quadrants. Cosine is always positive in the first and fourth quadrants.