Ch. 4.4

Example two: sec(-17pi/4)

Step 1. Find the reference angle, theta prime. -17pi/4 is -4 and 1/4 pi. less than -2pi. So add 6pi to find the coterminal angle. -17pi/4 + 6pi(24pi/4) = 7pi/4. 7pi/4 lies in quadrant 4, so 2pi(8pi/4)-7pi/4= pi/4. The function value of sec pi/4 is the square root of 2. Step 2. Use the quadrant angle in which theta prime lies to prefix the appropriate sign to the function value in step 1. The coterminal angle theta prime=7pi/4 which lies in quadrant 4. Since sec is positive in quadrant 4, we make square root of 2, positive.

Sec theta

r/x

Example: sin 135 degrees

Step 1. Find the reference angle, theta prime. Since 135 lies in quadrant two, 180-135=45 degrees. Function value: Sin 45 degrees is the square root of 2/2. Step Two. Use the quadrant in which theta prime lies to prefix the appropriate sign to the function value in step 1. So, 135 degrees lies in quadrant two. It's asking for sin of 135 degrees, and since sin is positive in quadrant two, it will be +square root of 2/2.

Example Three: Theta=580 degrees

To find the quadrant it lies in, 580-360=220 degrees. 220 degrees lies in quadrant 3. To find the reference angle, 220-180=40 degrees.

Procedure to for using reference angles to evaluate trig functions

1. find the reference angle, theta prime, and the function value of theta prime 2. use the quadrant in which theta prime lies in to find the appropriate sign of the function value in step 1

Example: Theta=345 degrees

345 degrees lies in quadrant four. The reference angle Theta Prime=360-345=15 degrees

Example Two: Theta=5pi/6

5pi/6 lies in quadrant two because 6pi/6 is pi and 3pi/6 is pi/2, so it lies in quadrant two. The reference angle is Theta Prime=pi-5pi/6 which equals 6pi/6-5pi/6=pi/6. So the reference angle of 5pi/6 is pi/6.

Signs of Trig Functions

A Smart Trig Class

Example Four: Theta=8pi/3

To find the quadrant subtract 8pi/3-2pi. 8pi/3-6pi/3= 2pi/3. 2pi/3 lies in quadrant two. To find the reference angle subtract pi from 2pi/3. 3pi/3-2pi/3=pi/3. The reference angle is pi/3.

Csc theta

r/y

P=(X,Y) Distance Formula

r=square root of x^2 + y^2

Reference Angles

the positive acute angle, theta prime, formed by the terminal side of Theta and the x-axis

Cot Theta

x/y

Sin theta

y/r

Cos Theta

y/x

Tan theta

y/x