1 Trigonometric Functions of Any Angle & Polar Coordinates Sections 8.1, 8.2, 8.3, 21.10

2 Definitions of Trigonometric Functions of Any Angle Let  be an angle in standard position with (x, y) a point on the terminal side of  and Definitions of Trig Functions of Any Angle y x  (x, y) r

3 Since the radius is always positive (r > 0), the signs of the trig functions are dependent upon the signs of x and y. Therefore, we can determine the sign of the functions by knowing the quadrant in which the terminal side of the angle lies. The Signs of the Trig Functions

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5 Where each trig function is POSITIVE: A CT S “All Students Take Calculus” Translation: A = All 3 functions are positive in Quad 1 S= Sine function is positive in Quad 2 T= Tangent function is positive in Quad 3 C= Cosine function is positive in Quad 4 *In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tan is positive, but sine and cosine are negative;... **The reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, …

6 Determine if the following functions are positive or negative: Example sin 210° cos 320° cot (-135°) csc 500° tan 315°

7 Trig functions of Quadrantal Angles To find the sine, cosine, tangent, etc. of angles whose terminal side falls on one of the axes, we will use the circle. (0, r) (r, 0) (-r, 0) (0, -r) 00

8 Find the value of the six trig functions for Example (r, 0) (0, r) (-r, 0) (0, -r) 00 

9 The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles. Reference Angles Definition of Reference Angle Let  be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of  and the horizontal axis.

10 Example Find the reference angle for Solution y x  By sketching  in standard position, we see that it is a 3 rd quadrant angle. To find, you would subtract 180° from 225 °.

11 So what’s so great about reference angles? Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies. For example, 45° is the ref angleIn Quad 3, sin is negative

12 Example Give the exact value of the trig function (without using a calculator).

13 Now, of course you can simply use the calculator to find the value of the trig function of any angle and it will correctly return the answer with the correct sign. Remember: Make sure the Mode setting is set to the correct form of the angle: Radian or Degree To find the trig functions of the reciprocal functions (csc, sec, and cot), use the button.

14 Example Evaluate the trig function to four decimal places. Set Mode to Degree Enter:    324   

15 HOWEVER, it is very important to know how to use the reference angle when we are using the inverse trig functions on the calculator to find the angle because the calculator may not directly give you the angle you want. r -5 y x  (-12, -5) -12 Example: Find the value of  to the nearest 0.01°

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17 In general, for  in radians, A second way to measure angles is in radians. Radian Measure (Sect 8.3) Definition of Radian: One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.

18 Radian Measure

19 Radian Measure

20 Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by Example Convert from Degrees to Radians: 210º

21 Conversions Between Degrees and Radians Example a) Convert from radians to degrees: b) Convert from radians to degrees: 3.8

22 Conversions Between Degrees and Radians c) Convert from degrees to radians (exact): d) Convert from radians to degrees:

23 Conversions Between Degrees and Radians Again! e) Convert from degrees to radians ( to 3 decimal places ): f) Convert from radians to degrees ( to nearest tenth ): 1 rad

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25 Polar Coordinates (Sect 21.10) A point P in the polar coordinate system is represented by an ordered pair. If, then r is the distance of the point from the pole.  is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point.

26 Polar Coordinates If, then the point is located units on the ray that extends in the opposite direction of the terminal side of .

27 Example Plot the point P with polar coordinates

28 Example (r,  )  Polar axis Pole Plot the point with polar coordinates 4

29 Plotting Points Using Polar Coordinates

30 Plotting Points Using Polar Coordinates

31 A)B) C) D)

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33 To find the rectangular coordinates for a point given its polar coordinates, we can use the trig functions. Example

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35 Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions. Example: Find the polar coordinates for the point for which the rectangular coordinates are given: (5, 4) Express answer to three sig digits. (5, 4)

36 Conversion from Rectangular Coordinates to Polar Coordinates If P is a point with rectangular coordinates (x, y), the polar coordinates (r,  ) of P are given by P You need to consider the quadrant in which P lies in order to find the value of .

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39 The TI-84 calculator has handy conversion features built-in. Check out the ANGLE menu. 5: Returns value of r given rectangular coordinates (x, y) 6: Returns value of  given rectangular coordinates (x, y) 7: Returns value of x given polar coordinates (r,  ) 8: Returns value of y given polar coordinates (r,  ) Check the MODE for the appropriate setting for angle measure (degrees vs. radians).

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