Rotational Trigonometry: Trig at a Point Dr. Shildneck The Trigonometric Ratios We all know Soh-Cah-Toa. The three basic trigonometric functions are the sine, cosine and tangent. But, there are three more trig functions called the reciprocal functions. Recall from before, Soh-Cah-Toa gives us The reciprocal functions,
named the cotangent, secant, and cosecant, are Example 1 Given the following triangle, find the values of all six trigonometric ratios for . 13 5 12
Trigonometry of Any Angle Let be any angle in standard position and a point P(x, y) be a point on the terminal side of . Let r be the non-zero distance from the origin to P. How could you use this information to find the distance r? The PYTHAGOREAN THEOREM, of course! x y r
So r 2 x 2 y 2 P(x,y) r x2 y2 Note: Since x2 and y2 are always positive, the square root of their sum is always positive. Therefore, r is always positive. Trigonometry of Any Angle Furthermore, we can look at the right triangle created by the terminal side (r),
And the x-axis to determine the values of the six trigonometric functions of . Now, for that angle of reference, then: opposite y is the _________________ x is the _________________ adjacent (adj) x (opp) y
r (hyp) P(x,y) r is the _________________ hypotenuse y r x cos
r y tan x sin r y r sec x x cot
y csc Example 2 Let the point (-4, 5) be a point on the terminal side of an angle in standard position. Find the values of all six trigonometric functions for the angle. Example 3 Let P be a point on the terminal side of standard of in 4 sin
Quadrant IV, such that . 7 Find the values of all six trigonometric functions for the angle. The Trigonometric Values of the Quadrantal Angles Recall that the quadrantal angles are those that are made up of the positive and negative axes. The angles are multiples of 90 or
radians. 2 o To determine the trigonometric values of any of these angles, use the definition of the trig functions at specific points. 1) Choose ANY point on the axis that makes the terminal side of the angle. 2) Determine the distance to that point (r). 3) Plug in the correct values for x, y, and/or r in the formulas we discovered earlier. Example 4 Find the values of each of the following.
Lets call this point P(0, 5) sin90 1. What is r (the distance from (0,0) to P)? 5 2. What the x-value of the point? 0 3. What the y-value of the point? 5
4. How do you find the sine? y 5 sin90 1 r 5 We could pick ANY number on this part of the axis. Example 5 Find the values of each of the following. cos270
Example 6 Find the values of each of the following. tan180 Example 7 Find the values of each of the following. sin Example 8 Find the values of each of the following.
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