Writing Assignment #5

1. Cyclic Function/Trigonometric Function y=1.5sin(2/3x)+3.5. A cyclic function is a function that has a graph that could go on forever, a trigonometric function is one that may not go on forever.

1.1. 1.5 = amplitude: the maximum height of the graph minus the minimum height of the graph divided by 2. (basically, the midline of the graph)

1.2. 2/3 = the "b" changes the horizontal stretch/compression, its the number of cycles in 2π, using the equation b=2π/p. 2π is one cycle and p represents the period. the period is the length of one cycle. in this case, there is a period of 3π. So 2π/3π = 2/3.

2. Unit Circle: A unit circle is a circle with a radius of one (a unit radius). In trigonometry, the unit circle is centered at the origin. the x value (or the horizontal distance) is cosϑ and the y value (or the vertical distance) is sinϑ.

2.1. Degrees

2.1.1. Ex: 3π/4 - a degree is one way to plot all 360° on a unit circle (with 15° intervals). The way to turn this into a degree, us you multiply the radian by 180/π. This will give you 135°

2.2. Radians

2.2.1. Ex: 135° - a radian is equal to 180/π. The radius of a unit circle (the cos value) joined with the vertical distance of the arc from zero (the sine value) create an angle and that angle is 1 radian.

3. Sine/Cosine/Tangent

3.1. Sine - sine is the vertical distance whether it be going up (positive) or going down (negative). to solve for sine, you use the equation Soh. which translates to Sineϑ=Opposite/Hypotenuse. The opposite and hypotenuse are sides of a triangle. once you find sine, (and hypotenuse was already given) you can use the pythagorean theory to solve for the hypotenuse of a triangle. (a^2 + b^2 = c^2).

3.2. Cosine - cosine is the horizontal distance whether it be going left (negative) or right (positive). to solve for cosine, you use the equation Cah. which translates to Cosϑ=adjacent/hypotenuse. once you find cosine, (and the hypotenuse was already given) you can use the pythagorean theory to solve for the sine (vertical distance) of a triangle and use that to find the points on a unit circle.

3.2.1. like (x,y) points on a graph, a unit circle has (cosϑ,sinϑ) points on a unit circle. After using the Soh or Cah to find the sineϑ or cosϑ, you will result in points like (1/2, sqrt(3)/2)

3.3. Tangent - Tangent is the hypotenuse of a triangle (or the angled distance on a unit circle). Tanϑ=sinϑ/cosϑ. So given two variables, you can use this equation (as well as the ones listed above) to solve for the missing variable. You will also notice that Toa (in the SohCahToa rule) is Tanϑ=opposite/adjacent. you can assume that opposite = sinϑ and adjacent = cosϑ

4. Law of Sine and Cosine

4.1. Sine - the ratio of the sine of an angle to the length of the side opposite the angle is constant. that means that sinA/a=sinB/b=sinC/c. This property solves for measures of angles and lengths of sides for any triangle, not just a right angle.

4.2. Cosine - Just like the Law of sines, the law of cosine represents a relationship between the sides and angles of a triangle. Specifically, when given the lengths of any two sides, such as a and b, and the angle between them, C, the length of the third side can be found using the relationship: c^2 = a^2 + b^2 - 2ab cos(C)

5. Graphing Sine/Cosine/Tangent: a graph mimics the shape of a unit circle except on an (x,y) coordinate graph. as opposed to having the points revolute around a circle, this graph movies infinitely outwards.

5.1. Sine vs. Cosine: The general form of the cosine function can also be y=asin(b(x-c))+D. since the cosine function is identical to the sine function (except for a horizontal shifted to the left of 90º or π/2 radians). Any cosine function can be written as a sine function as: cos(x)=sin(x+π/2)

5.2. Tan: Using the rule tanϑ=sinϑ/cosϑ , the tangent will be undefined wherever its denominator (the cosine) is zero. A zero in the denominator means you'll have a vertical asymptote. So the tangent will have vertical asymptotes wherever the cosine is zero: at –π/2, π/2, and 3π/2.