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Understanding Trigonometric Functions: Degrees vs. Radians Explained

Explore the difference between using real numbers, radians, and degrees in trigonometric functions. Learn how the unit circle defines sine, cosine, tangent, and more. Solve trigonometric function values for specific angles.

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Understanding Trigonometric Functions: Degrees vs. Radians Explained

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  1. 6.3.1 Trigonometric Functions of Real Numbers

  2. Radians vs. Real Numbers • The argument of a trig function can be a real number, radians, or degrees. • Sin(2) real number, radian, or degree? • Sin(2) real number, radian, or degree? • Note: • sin(2) ≠ sin(2)

  3. Unit Circle • The unit circle defines the trig functions in terms of the dependent variable • If we consider time in radians as our x-values we can consider the yt plane (similar to xy plane except t values for x axis)

  4. sin(t) = y cos(t) = x tan(t) = csc(t) = sec(t) = cot(t) = Functions definedP(x,y) on the unit circle

  5. Since these functions are defined by a circle… • On a circle 180 or half a circle or  • Produces a t-value of exact opposite value • Similarly 360 or a whole circle or 2 • Produces a t-value of the same value

  6. This defines our functions as follows: • P(t) = (x, y) • P(t + ) = (-x, -y) • P(t - ) = (-x, -y) • P(-t) = (x, -y)

  7. Given, Since P(-t - ) = P(-(t + )) Find P(t +  ), P(t -  ), P(-t), and P(-t - ) P(-t - ) P(t) P(-t) P(t + ), P(t - )

  8. Remember! • P(x,y) = P(cos(t) , sin(t)) • Where t is the angle in radians

  9. 10)a) - sin(-)= 0 cos(-)= -1 tan(-)= 0 csc(-)= U sec(-)= -1 cot(-)= U Find the Values of the Trigonometric Fucntions  0 (2) (-1, 0)

  10. 10)b) 6 sin(6)= cos(6)= tan(6)= csc(6)= sec(6)= cot(6)= You Try!  0 (2)

  11. Homework • p. 436 1-4, 5-8, 9-15 odd

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