Unit Circle

Unit Circle

What Is the Unit Circle?

What is the unit circle?

• It is a circle centered at the origin (0,0) with a radius of 1

• It has the formula x² + y² = 1

• It is a shortcut to finding the sine and cosine of the angles of special right triangles (30-60-90) or (45-45-90)

Why does the unit circle work?

• The distance from the center to a point on the circle is the radius, or 1.

• If we drop a perpendicular line from the point to the radius, that vertical distance is y

• If we draw a line from the center to where the perpendicular line touches the circle, that horizontal distance is x

Why does the unit circle work?

• Remember our trig functions (SOHCAHTOA)?

•  $\sin\ \left(\Theta\right)\ =\ \frac{opp}{hyp}\ \ ,\ \ \cos\ \left(\Theta\right)\ =\ \frac{adj}{hyp}$  

•  $\tan\ \left(\Theta\right)\ =\ \frac{opp}{adj}$  

Why does the unit circle work?

• sin ( $\Theta$  ) = opp/hyp = y/1 = y

• cos ( $\Theta$  ) = adj/hyp = x/1 = x

• coordinate of a point on the circle (x, y) is (sin ( $\Theta$  ), cos (  $\Theta$ ) ) 

• substitute sin ( $\Theta$  ) for x, and      cos ( $\Theta$ )  for y

How do you divide the unit circle?

Division 1

• Divide the circle into four equal sections

• The sections represent 0^o, 90^o, 180^o, and 360^o (360^o = 0^o)

• To change from degrees to radians you multiply the degrees by  $\frac{\pi}{180}$  

• In radians, these divisions are  $0,\ \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ 2\pi$  

Division 2

• Divide each quarter into thirds

• This will represent the angles 30, 60, (90), 120, 150, (180), 210, 240 (270), 300, 330 (360)

• If you convert these to radians, you have  $0,\frac{\pi}{6},\ \frac{\pi}{3},\ \frac{\pi}{2},\ \frac{2\pi}{3},\ \frac{5\pi}{6},\ \pi$  

•  $\frac{7\pi}{6},\ \frac{4\pi}{3},\ \frac{3\pi}{2},\frac{5\pi}{3},\ \frac{11\pi}{6},\ 2\pi$  

Division 3

• Divide the 90^o angles in 1/2, to make 45^o angles

• This adds angles  $45^o,\ 135^o,\ 225^o,\ 315^o$  

• or  $\frac{\pi}{4},\ \frac{3\pi}{4},\ \frac{5\pi}{4,},\ \frac{7\pi}{4}$  

What are the values for sine and cosine for each angle on the unit circle?

Values for Cosine and Sine

• The cosine and sine of each angle are the (x, y) values.

• Cosine is the x value

• Sine is the y-value

• The signs correspond to the sign of the quadrants of the coordinate plane

Values for the Cosine and Sine

• The values for the cosine and sine correspond to the values of the cosine and sign of those special angles (30 - 60 -90), (45-45-90)

• There is a trick

• Starting with 0o, label that point (1,0), where 1 is the radius.

• As you move up to 30, 45, 60, the cosine will be  $\frac{\sqrt{3}}{2},\ \frac{\sqrt{2}}{2},\frac{\sqrt{1}}{2}$  

• As you move up to 30, 45, 60, the sine will be just the opposite:  $\frac{\sqrt{1}}{2},\ \frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2}$  

A Couple of Things to Notice

• Everything has 2 in the denominator

• Every numerator has a radical, but the  $\sqrt{1}$  is 1, so it is not written as a radical.

• The cosines go in descending order moving from 0 to 90                    (  $\sqrt{3},\sqrt{2},\ \sqrt{1}$  ) and the sines go in ascending order                        ( $\sqrt{1},\ \sqrt{2},\ \sqrt{3}$  )

• You could go up with the cosines in 3-2-1 order, and then come back down from 60 in 1 -2 -3 order

• 0, 90, 180, 270 will have coordinates (1,0), (0, 1), (-1,0) , (0, -1) respectively. These are the coordinates of the radii.

Values for Cosine and Sine

• Start at 180^o, moving up to 90^o, and repeat the same pattern EXCEPT, x-values are negative in the second quadrant shich means cosine for angles between 90 and 180 will have negative signs.

• Start at 180^o and move down to 270^o, repeat the same pattern EXCEPT both x and y (cosine and sine) are negative, because both x and y are negative in Q-III

Values for Cosine and Sine

• Go to 0, and move down toward 270^o, repeating the same pattern EXCEPT the x-value will be + and the y value will be (-), or the cosine is + and the sine is (-). X is positive and y is negative in QIV.

• 
  

Remember

• Up from 0 to 90, and down from 0 to 270, the pattern is the same except for the signs:  $\sqrt{3},\ \sqrt{2},\ \sqrt{1}$  for x (cosine), and  $\sqrt{1},\ \sqrt{2},\ \sqrt{3}$  for y (sine)  (all with a denominator of 2)

• Up from 180 to 90, and down from 180 to 270, the pattern is the same except for the signs:  $\sqrt{3},\ \sqrt{2},\ \sqrt{1}$  for x (cosine) and  $\sqrt{1},\ \sqrt{2},\ \sqrt{3}$  for y (sine) (all with a denominator of 2.)

• Signs are the signs of the quadrants: QI (+x, +y),  QII (-x, +y), QIII (-x, -y) and QIV (+x, -y)

• To find the radian equivalent of the angle, multiple the degrees by  $\frac{\pi}{180}$ . To find the degree equivalent of a radian, multiply by  $\frac{180}{\pi}$  

Multiple Choice

At what point are sine and cosine the same value?

• 0

• 90

• 45

• 60

Multiple Choice

What is the value of sine at 330

• 1/2

• -1/2

• √3 / 2

• -√3 / 2

Multiple Choice

What is the exact coordinates of 3π/4 on the unit circle?

(Multiply by  $\frac{\pi}{180}$  to get degrees)

• (−√2∕2, √2∕2)

• (−√2∕2, −√2∕2)

• (−√3∕2, −√2∕2)

• (√2∕2, −√2∕2)

Multiple Choice

sin π/4

(Multiply by   $\frac{180}{\pi}$  )

• √2∕2

• 1/2

• √3/2

• 0

Multiple Choice

cosθ is positive in

• the 1st and 2nd quadrants
• the 1st and 3rd quadrants
• the 1st and 4th quadrants
• the 2nd and 3rd quadrants

Multiple Choice

What is the radius of a unit circle?

• 1
• 2
• 1/2
• 0

Multiple Choice

sinθ is positive in

• the 1st and 2nd quadrants
• the 1st and 3rd quadrants
• the 1st and 4th quadrants
• the 2nd and 3rd quadrants

Multiple Choice

sin 7π/6

(Multiply by  $\frac{180}{\pi}$  )

• 1/2

• -1/2

• √3/2

• -√3/2

Multiple Choice

cos(60°)

• 0

• 1/2

• √3/2

• 1

Multiple Choice

cos(120°)

• 0

• 1/2

• √3/2

• 1