What is the unit circle?

• The unit circle is a circle with radius 1 and centre (0, 0)
• Angles are always measured from the positive x-axis and turn:
  • anticlockwise?for?positive?angles
  • clockwise?for?negative?angles
• It can be used to calculate trig values as a coordinate point (x, y) on the circle
  • Trig values can be found by making a right triangle with the radius as the hypotenuse
  • θ is the angle measured anticlockwise from the positive?x-axis
  • The?x-axis will always be adjacent to the angle,?θ
• SOHCAHTOA?can be used to find the values of sinθ, cosθ?and tanθ?easily
• As the radius is 1 unit
  • the?x?coordinate?gives the value of?cosθ
  • the?y?coordinate?gives the value of?sinθ
• As the origin is one of the end points - dividing the y coordinate by the x coordinate gives the gradient
  • the?gradient?of the line gives the value of?tanθ

How is the unit circle used to construct the graphs of sine and cosine?

• On the unit circle the?y-coordinates?give the value of?sine
  • Plot the?y-coordinate from the unit circle as the?y-coordinate on a trig graph for?x-coordinates of?θ?= 0, π/2, π, 3π/2 and 2π
  • Join these points up using a smooth curve
    • To get a clearer idea of the shape of the curve the points for?x-coordinates of?θ?= π/4, 3π/4, 5π/4 and 7π/4 could also be plotted

• On the unit circle the?x-coordinates?give the value of?cosine
  • Plot the?x-coordinate from the unit circle as the?y-coordinate on a trig graph for?x-coordinates of?θ?= 0, π/4, π/2, 3π/4 and 2π
  • Join these points up using a smooth curve
    • To get a clearer idea of the shape of the curve the points for?x-coordinates of?θ?= π/4, 3π/4, 5π/4 and 7π/4 could also be plotted

• Looking at the unit circle alongside of the sine or cosine graph will help to visualise this clearer

What are the properties of the unit circle?

• • • Cosθ is positive in this quadrant
  • Starting from the?fourth?quadrant (on the bottom right) and working anti-clockwise the positive trig functions spell out?CAST
    • This is why it is often thought of as the?CAST?diagram
    • You may have your own way of remembering this
    • A popular one starting from the first quadrant is?All Students Take Calculus
  • To help picture this better try sketching all three trig graphs on one set of axes and look at which graphs are positive in each 90° section

How is the unit circle used to find secondary solutions?

• Trigonometric functions have more than one input to each output
  • For example sin 30° = sin 150° = 0.5
  • This means that trigonometric equations have more than one solution
  • For example both 30° and 150° satisfy the equation sin?x?= 0.5
• The unit circle can be used to find all solutions to trigonometric equations in a given interval
  • Your calculator will only give you the first solution to a problem such as?x?= sin^-1(0.5)
    • This solution is called the?primary value
  • However, due to the?periodic?nature of the trig functions there could be an infinite number of solutions
    • Further solutions are called the?secondary values
  • This is why you will be given a?domain?in which your solutions should be found
    • This could either be in degrees or in radians
    • If you see π or some multiple of π then you must work in radians
• The following steps may help you use the unit circle to find?secondary values

STEP 1: Draw the angle into the first quadrant using the?x?or?y?coordinate to help you

• If you are working with sin?x?=?k, draw the line from the origin to the circumference of the circle at the point where the?y coordinate?is?k
• If you are working with cos?x?=?k, draw the line from the origin to the circumference of the circle at the point where the?x coordinate?is?k
• If you are working with tan?x?=?k, draw the line from the origin to the circumference of the circle such that the gradient of the line is?k
  • This will give you the angle which should be measured from the?positive x-axis…
    • … anticlockwise for a positive angle
    • … clockwise for a negative angle

STEP 2: Draw the radius in the other quadrant which has the same...

• ... x-coordinate ifsolving cos?x?=?k
  • This will be the quadrant which is vertical to the original quadrant
• ... y-coordinate ifsolving sin?x?= k
  • This will be the quadrant which is horizontal to the original quadrant
• ... gradient if solving tan x = k
  • This will be the quadrant diagonal to the original quadrant

STEP 3: Work out the size of the second angle, measuring from the positive x-axis

• … anticlockwise for a positive angle
• … clockwise for a negative angle
  • You should look at the given range of values to decide whether you need the negative or positive angle

STEP 4: Add or subtract either 360° or 2π?radians to both values until you have all solutions in the required range