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6.3 Trig Functions of Angles

6.3 Trig Functions of Angles. So P = (x, y) is a point on the terminal side of theta. Using the P ythagorean theorem, r = . P. P (x, y). opposite. r. y. hypotenuse. Q. O. Q. x. adjacent. O. So:.

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6.3 Trig Functions of Angles

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  1. 6.3 Trig Functions of Angles

  2. So P = (x, y) is a point on the terminal side of theta. Using the Pythagorean theorem, r = P P (x, y) opposite r y hypotenuse Q O Q x adjacent O

  3. So:

  4. ExLet (-8, -6) be a point on the terminal side of theta. Find all six trig functions of theta.

  5. Quadrantal Angels – angels that are coterminal with the coordinate axis. • When the terminal side of an angle theta, that is in standard position , lies on the coordinate axes. (The angles for which either the x- or y-coordinate of a point on the terminal side of the angle is 0.) • Since division by 0 is undefined, certain trig functions are not defined for certain angles. EX is tan 90º degrees = y/x because x = 0.

  6. Evaluating Trig Functions at Any Angle • ‘r’ is always positive because it is the distance from the origin to the point P(x, y). The position of the terminal side will determine whether the trig function is positive or negative. • Q I: all positive • Q2: (-x, y): sin, csc are + • Q3 (-x, -y): tan, cot are + • Q4 (-x, -y): cos, sec are +

  7. EX • sin (-180º)

  8. We see that the trig functions for angles that aren’t acute have the same value except possibly for the sign, as the corresponding trig functions of an acute angle. Let theta be an angle in standard position. The reference angle (theta bar) associated with theta is the acute angle formed by the terminal side of theta and the x-axis.

  9. ExFind the reference angle for:

  10. ExFind the following: Cos 135º = tan 390º cos 60º tan 240º

  11. Evaluating Trig Functions for any angle: • Find the reference angle (theta bar) associated with the angle theta. • Determine the sign of the trig function of theta by noting the quadrant in which theta lies. • The value of the trig function of theta is the same, except possibly for the sign, as the value of the trig function theta bar.

  12. Evaluate the Trig Functions using the Reference Angle.

  13. Fundamental Identities

  14. Ex express sin in terms of cos

  15. Exexpress tan in terms of sin

  16. Ex

  17. Ex

  18. Ex

  19. Area of Triangles: With sides of length a & b and with included angle theta. • Find the area of triangle ABC: C 3cm 120º B A 18cm

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