Unit Circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.If (x, y) is a point on the unit circle, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equationx^2 + y^2 = 1.

Math Joke of the week:

Q: Why didn't sin and tan go to the party?
A: Just cos

Chapter 4 summary

 Relating the sides of triangles to the unit circle, which accounts for special triangles and angles. The things to remember are (cos,sin) = (x,y), and cosine is equivalent to adjacent side over hypotenuse, whereas sine is opposite over hypotenuse and tangent is opposite over adjacent. This information allows us to find the points along the unit circle, finding the coordinates along the circumference using 45-45-90 and 30-60-90 right triangles sine and cosine values. Also, we learned that trigonometry has many identities, which are tools used for trigonometric equations to solve. We can prove the existence of these identities using verifying. Inverses are for solving for arcsin1/2 and we are solving for x. We solve by taking the sin^-1 of 1/2 and that is x.

Math Joke of the Week:

Q: What do you get when you cross a mosquito with a mountain climber?
A: Nothing. You can't cross a vector and a scalar.

Trigonometric Equations

Solving trig equations use both the reference angles you've memorized and a lot of the algebra learned.

Solutions of trigonometric equations may also be found by examining the sign of the trig value and determining the proper quadrant(s) for that value. 

Be prepared to solve sin(x) + 2 = 3 for 0° < x < 360°

Just as with linear equations, I'll first isolate the variable-containing term:   sin(x) + 2 = 3   sin(x) = 1

Now use the reference angles memorized:   x = 90°

Solve tan2(x) + 3 = 0 for 0° < x < 360°

There's the temptation to quickly recall that the tangent of 60° involves the square root of 3 and slap down an answer, but this equation doesn't actually have a solution:tan2(x) = –3

How can the square of a trig function evaluate to a negative number? It can't, no solution

Math Joke of the week:

Q: What did one Calculus book say to the other?
A: Don't bother me I've got my own problems!

Solving Trig Identities 

Proving an identity is very different in concept from solving an equation. Though you'll use many of the same techniques, they are not the same, and the differences are what can cause you problems.

Always start off with the bigger or harder side of the identities.An "identity" is an equation or statement that is always true, no matter what. For instance, sin(x) = 1/csc(x) is an identity. To "prove" an identity, you have to use logical steps to show that one side of the equation can be transformed into the other side of the equation. You do not plug values into the identity to "prove" anything. Identities such as: sin2q+cos2q=1 

tan2 q +1 = sec2 q 

1+cot2q =csc2q

are needed to use to solve the identities 

 

Math Joke of the week

Q: Why is beer never served at a math party?
A: Because you can't drink and derive.