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Given an ordered pair, calculate the 6 trigonometric ratios

I chose this topic because I struggled with learning the trig functions as a whole. I especially was confused with which was the reciprocal for each. Looking back on the topic and using outside sources to re-teach it to myself, it's very easy to understand. First, we must recap the trigonometric ratios. Previously, we used these ratios to solve triangles. The basic three we know from geometry include:
 * 1) Tangent
 * 2) Cosine
 * 3) Sine

Then, we must learn the new trigonometric ratios.
 * 1) Cotangent
 * 2) Secant
 * 3) Cosecant

These new trigonometric ratios are reciprocals of the trigonometric ratios we learned in geometry.
 * Cotangent is the reciprocal of tangent
 * Secant is the reciprocal or Cosine
 * Cosecant is the reciprocal of Sine

Each of these trigonometric relations has their own ratio, based on the unit circle. The ** unit circle ** is simply a circle that lies on the coordinate plane with its center located at the origin and a radius of 1.

Given an ordered pair, you can calculate the six trigonometric functions. The ordered pairs come from the unit circle. Different spots (angles from standard position) on the unit circle have different coordinates. For example:



Each ratio goes as follows:
 * 1) tan: y-coordinate/x-coordinate
 * 2) cos: x-coordinate/radius (the radius on a unit circle is always 1!)
 * 3) sin: y-coordinate/radius
 * 4) cot: x-coordinate/y-coordinate
 * 5) sec: radius/x-coordinate
 * 6) csc: radius/y-coordinate

The video will walk you through the steps completely: [|trig ratios.mov]


 * Note, making this powerpoint into a "movie" was the only way it would fit on this page. To hear sound, pause each slide and press the sound button.

Not only can you use the trig ratios with the unit circle, but you can use them with other x and y coordinates. There are still x and y values since coordinates are written in the form of (x, y). The coordinate (x, y) is equal to (cos, sin). But then, where does r come from? In order to find r you must use the pythagorean theorem, a^2 + b^2 = c^2, or in this case, x^2 + y^2 = r^2. So, r is equal to the square root of (x^2 + y^2). So, for example, let's take the coordinate (3, 4). 3^2 + 4^2 = 25. The square root of 25 is 5. 5 is the r value that you would substitute in. For example, the cosine ratio would be (x/r) or in this case (3/5).

Do not forget how to calculate the 6 trigonometric ratios!

1) cos = x/r 2) sin = y/r 3) tan = y/x 4) sec = r/x 5) csc = r/y 6) cot = x/y Now, try some of your own. Given the ordered pair, calculate the 6 trigonometric ratios:

a) Ordered pair for 45 degrees

b) Ordered pair for 120 degrees

c) (12, 5)

d) (7, 24)

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