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Unit Circle Trigonometry

Pre-Calculus Day 39. Unit Circle Trigonometry. What You Should Learn. Identify a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Plan for the day…. Review of homework Quiz Quick Review Let’s build the Unit Circle…

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Unit Circle Trigonometry

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  1. Pre-Calculus Day 39 Unit CircleTrigonometry

  2. What You Should Learn Identify a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle.

  3. Plan for the day… • Review of homework • Quiz • Quick Review • Let’s build the Unit Circle… • Homework

  4. Trigonometric Functions θ adj opp sin  = cos  = tan  = csc  = sec  = cot  = hyp adj hyp hyp adj opp adj opp hyp The trigonometric functions are opp adj sine, cosine, tangent, cotangent, secant, and cosecant. Note: sine and cosecant are reciprocals, cosine and secant are reciprocals and tangent and cotangent are reciprocals

  5. Fundamental Trigonometric Identities Fundamental Trigonometric Identities for Cofunction Identities sin  = cos(90  ) cos  = sin(90  ) sin  = cos (π/2  ) cos  = sin(π/2  ) tan  = cot(90  ) cot  = tan(90  ) tan  = cot(π/2  ) cot  = tan(π/2  ) sec  = csc(90  ) csc  = sec(90  ) sec  = csc(π/2  ) csc  = sec(π/2  ) Reciprocal Identities sin  = 1/csc cos = 1/sec tan = 1/cotcot = 1/tan sec = 1/cos csc = 1/sin Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2 + 1 = sec2 cot2  + 1 = csc2 

  6. Name the Equivalent Function…

  7. sin θ 1/csc

  8. tan θ sin θ/cos

  9. sec θ csc (90o-)

  10. cot θ cos θ/sin

  11. tan θ 1/cot

  12. cos θ sin (90o -)

  13. sec θ 1/cos

  14. A circle defined by x2 + y2 = 1 Imagine the real number line wrapped around the circle. Each real number t corresponds to a point (x, y). Since the radius is 1, the number t would correspond with the central angle (s = rθ). The Unit Circle (0, 1) (x, y) t (-1, 0) (1, 0) θ= t (0, -1)

  15. 45 1 45 Geometry of the 45-45-90 triangle Consider an isosceles right triangle with a hypotenuse the length of 1. What would be the length of the sides?

  16. Example: Trig Functions for  30 1 60○ 30 Geometry of the 30-60-90 triangle Consider a 30-60-90 triangle with a hypotenuse the length of 1. What would be the length of the sides?

  17. The Unit Circle • The values on the unit circle are obtained by overlaying the special right triangles in each quadrant. • Because the hypotenuse of each triangle has a length of 1, and starts at the origin, then the coordinates of the endpoint of the hypotenuse correlate with the lengths of the adjacent and opposite sides. • Likewise due to the hypotenuse length of 1, the cos and sin of each angle reduces to adj and hyp, respectively. • Therefore, the coordinates of the endpoint of the hypotenuse are (cos, sin ).

  18. Definitions of Trig Functions As mentioned, a real number t will correspond to a point (x, y) on the unit circle, and the angle which forms that arc will have a measure of t radians.

  19. Try this Evaluate the six trigonometric functions at each real number: a) t = π/6 b) t = 5π/4 c) t = 0 d) t = π e) t = - π/6 f) t = 13π/6

  20. Domain of Sine and Cosine • The domain of the sine and cosine functions is the set of all real numbers. • To find the range, consider the unit circle. Remember that sin t = y and cos t = x. • The range for sin and cos is [-1, 1].

  21. Periodic Functions • A function is periodic if there exists a positive real number c such that:for all t in the domain of f. • The smallest number c for which f is periodic is called the period of f. • Because the outputs of sin and cos repeat every full rotation, the period for these functions is 2π (or 360).

  22. Even and Odd Trig Functions • Cosine and secant functions are even cos (-t) = cos t sec (-t) = sec t • Sine, cosecant, tangent and cotangent are odd sin (-t) = - sin t csc (-t) = - csc t tan (-t) = - tan t cot (-t) = - cot t

  23. Homework 21 Section 4.2 pp. 278-279 # 1-39 odd, 45, 49 Memorize your unit circle!!! Quiz after break!!

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