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THE UNIT CIRCLE

THE UNIT CIRCLE. Definitions of Trigonometric functions. Let t be a real number and let (x,y) be the point on the unit circle corresponding to t Sin t = y csc t = 1/y Cos t = x sec t = 1/x Tan t = y/x cot t = x/y.

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THE UNIT CIRCLE

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  1. THE UNIT CIRCLE

  2. Definitions of Trigonometric functions • Let t be a real number and let (x,y) be the point on the unit circle corresponding to t • Sin t = y csc t = 1/y • Cos t = x sec t = 1/x • Tan t = y/x cot t = x/y

  3. A circle with center at (0, 0) and radius 1 is called a unit circle. The equation of this circle would be (0,1) (-1,0) (1,0) (0,-1) So points on this circle must satisfy this equation.

  4. Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value? You can see there are two y values. They can be found by putting 1/2 into the equation for x and solving for y. x = 1/2 (0,1) (-1,0) (1,0) We'll look at a larger version of this and make a right triangle. (0,-1)

  5. We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always 1 because it is a radius of the circle. (0,1) (-1,0) (1,0)  (0,-1) Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

  6. You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly.

  7. Let’s think about the function f() = sin  What is the domain? (remember domain means the “legal” things you can put in for  ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1  sin   1 (0, 1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for sine? (sine is the y value so what is the lowest and highest y value?) (1, 0) (-1, 0) (0, -1)

  8. Let’s think about the function f() = cos  What is the domain? (remember domain means the “legal” things you can put in for  ). You can put in anything you want so the domain is all real numbers. What is the range? (remember range means what you get out of the function). The range is: -1  cos   1 (0, 1) Let’s look at the unit circle to answer that. What is the lowest and highest value you’d ever get for cosine? (cosine is the x value so what is the lowest and highest x value?) (-1, 0) (1, 0) (0, -1)

  9. Let’s think about the function f() = tan  What is the domain? (remember domain means the “legal” things you can put in for  ). Tangent is y/x so we will have an “illegal” if x is 0. x is 0 at 90° (or /2) or any odd multiple of 90° The domain then is all real numbers except odd multiples of 90° or  /2. What is the range? (remember range means what you get out of the function). If we take any y/x, we could end up getting any value so range is all real numbers.

  10. Let’s think about the function f() = csc  What is the domain? Since this is 1/sin , we’ll have trouble if sin  = 0. That will happen at 0 and multiples of  (or 180°). The domain then is all real numbers except multiples of . Since the range is: -1  sin   1, sine will be fractions less than one. If you take their reciprocal you will get things greater than 1. The range then is all real numbers greater than or equal to 1 or all real numbers less than or equal to -1. What is the range?

  11. Let’s think about the function f() = sec  What is the domain? Since this is 1/cos , we’ll have trouble if cos  = 0. That will happen at odd multiples of /2 (or 90°). The domain then is all real numbers except odd multiples of /2. Since the range is: -1  cos   1, cosine will be fractions less than one. If you take their reciprocal you will get things greater than 1. The range then is all real numbers greater than or equal to 1 or all real numbers less than or equal to -1. What is the range?

  12. Let’s think about the function f() = cot  What is the domain? Since this is cos /sin , we’ll have trouble if sin  = 0. That will happen at 0 and multiples of  (or 180°). The domain then is all real numbers except multiples of . What is the range? Like the tangent, the range will be all real numbers. The domains and ranges of the trig functions are summarized in your book in Table 6 on page 542. You need to know these. If you know the unit circle, you can figure these out.

  13. This would have the same value as Reciprocal functions have the same period. PERIODIC PROPERTIES sin( + 2) = sin cosec( + 2) = cosec  cos( + 2) = cos  sec( + 2) = sec  tan( + ) = tan  cot( + ) = cot  1 (you can count around on unit circle or subtract the period twice.)

  14. Now let’s look at the unit circle to compare trig functions of positive vs. negative angles. Remember negative angle means to go clockwise

  15. If a function is even, its reciprocal function will be also. If a function is odd its reciprocal will be also. EVEN-ODD PROPERTIES sin(-  ) = - sin (odd)cosec(-  ) = - cosec (odd) cos(-  ) = cos  (even) sec(-  ) = sec (even) tan(-  ) = - tan (odd) cot(-  ) = - cot (odd)

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