§5.3

 I can use the definitions of trigonometric functions of any angle.  I can use the signs of the trigonometric functions.  I can find the reference angles.  I can use the reference angles to evaluate trigonometric functions.

 Up until this point, all angles that we have looked at have been acute angles. The trig functions of other angles can be evaluated as well.

 Note: (a) is in the first quadrant and we can use our right triangle trig functions.  Note: (b), (c), and (d) are shown in standard position, but are not acute.  We can extend our definitions of the trig functions to include such angles and quadrantal angles.

Let θ be any angle in standard position and let P=(x, y) be a point on the terminal side on θ. If is the distance from(0, 0) to (x, y) the six trig functions are defined as: Why did we not have to set r ≠ 0? Distances are always positive!

Let P=(-3, -5) be a point on the terminal side of θ. Find each one of the six trig functions. Fir st find r:

2. Let P=(1, -3) be a point on the terminal side of θ. Find each one of the six trig functions.

Evaluate, if possible, the sine and tangent functions at the following four quadrantal angles. a. θ = 0˚ b. θ = 90˚ c. θ = 180˚ d. θ= 270˚ Solutions: θ = 0˚ Use:

If Θ is not a quadrantal angle, the sign of a trig function depends on the quadrant in which Θ lies.

 If tanθ 0, name the quadrant in which angle θ lies.  If sinθ < 0 and cosθ < 0, name the quadrant in which angle θ lies. Example 3

 Given tanθ = -2/3 and cosθ > 0, find cosθ and cscθ. Example 4 Because tangent is neg. and cosine is positive, θ lies in quadrant IV In quadrant IV x is positive and y is negative Thus, x= 3 and y = -2 Now we can find r And use that to find cos θ and csc θ

 Given tanθ = -1/3 and cosθ < 0, find sinθ and secθ.

 Definition of a Reference Angle Let θ be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis. Find the reference angle, θ’, for each of the following angles: a. θ = 345° b. θ = c. θ= -135˚ d. θ= 2.5

a. θ= 345˚ b. θ= Because 345˚ is I quadrant IV the reference angle is: Notice: no degree symbol, so this must be in radians Because is between and the reference angle is in quadrant II, the reference angle is:

c. θ= -135º d. θ= 2.5 Because -135˚ in in quadrant III the reference angle is: Notice: no degree symbol, so this must be in radians Because 2.5 radians is between and in quadrant II the reference angle is:

1. Find a positive angle less than 360° that is coterminal with the given angle. 2. Draw in standard position. 3. Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side and the x-axis is the reference angle.

a. θ= 580º b. θ= c. θ= a. θ= 580º Subtract 360º Draw the coterminal angle (α) Use the drawing to find the reference angle.

b.c. So subtract 2π Draw the angle For this angle I would draw the angle first. That will be the easiest way to find the positive reference angle.

 The values of the trig functions of a given angle, θ, are the same as the values of the trig functions of the reference angle, θ’, except possibly for the sign.  A function value of the acute reference angle, θ’, is always positive.  However, the same function value for θ may be positive or negative.

a. b. c. d. e. f. Hints: 1. Draw the angle 2. Think of the trig definitions & properties 3. Use your unit circle!

a. b.

How did they get

Draw the angle to get the reference angle. How did they get

Draw the angle to get the reference angle: Use: Answer: