Trigonometric Functions of Any Angle & Polar Coordinates MAT 205 SPRING 2009 Sections 8.1, 8.2, 8.3, 21.10 Trigonometric Functions of Any Angle & Polar Coordinates

Definitions of Trigonometric Functions of Any Angle Definitions of Trig Functions of Any Angle (Sect 8.1) 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 y x  (x, y) r

The Signs of the Trig Functions 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

A C T S Where each trig function is POSITIVE: “All Students Take Calculus” A C T S 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, tangent is positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and tangent are negative. **Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, and cotangent is positive wherever tangent is positive.

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

Examples For the given values, determine the quadrant(s) in which the terminal side of θ lies.

Examples Determine the quadrant in which the terminal side of θ lies, subject to both given conditions.

Examples Find the exact value of the six trigonometric functions of θ if the terminal side of θ passes through point (3, -5).

Reference Angles (Sect 8.2) 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. 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.

Example Find the reference angle for Solution  y MAT 205 SPRING 2009 Example Find the reference angle for Solution y x  By sketching  in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.

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, In Quad 3, sin is negative 45° is the ref angle

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

Examples (Text p 239 #6 & 8) Express the given trigonometric function in terms of the same function of a positive acute angle.

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 or enter [original function] .

Evaluate . Round appropriately. Example Evaluate . Round appropriately. Set Mode to Degree Enter:  OR 

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°

Examples

Examples

BONUS PROBLEM

SUPER DUPER BONUS PROBLEM

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. (1, 0) (-1, 0) (0, -1) 0 (0, 1) Unit Circle: Center (0, 0) radius = 1 x2 + y2 = 1

Now using the definitions of the trig functions with r = 1, we have:

Example Find the value of the six trig functions for (0, 1) (-1, 0) (1, 0) (0, 1) (-1, 0) (0, -1) 0 

Example Find the value of the six trig functions for MAT 205 SPRING 2009 Example Find the value of the six trig functions for

Example Find the value of the six trig functions for MAT 205 SPRING 2009 Example Find the value of the six trig functions for

Radian Measure (Sect 8.3) Definition of Radian: A second way to measure angles is in radians. 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. In general, for  in radians,

Radian Measure

Radian Measure

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

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

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

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

Examples

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.

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

Example Plot the point P with polar coordinates

MAT 205 SPRING 2009 Example Plot the point with polar coordinates 4

Plotting Points Using Polar Coordinates MAT 205 SPRING 2009 Plotting Points Using Polar Coordinates

Plotting Points Using Polar Coordinates

A) B) D) C)

MAT 205 SPRING 2009 To find the rectangular coordinates for a point given its polar coordinates, we can use the trig functions. Example

MAT 205 SPRING 2009 Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions. Example: Find the polar coordinates (r, θ) for the point for which the rectangular coordinates are (5, 4). Express r and θ (in radians) to three sig digits. (5, 4)

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 .

Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees.

Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees.

5: Returns value of r given rectangular coordinates (x, y) MAT 205 SPRING 2009 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).

MAT 205 SPRING 2009 End of Section