THE UNIT CIRCLE, THE CAST RULE, AND SPECIAL ANGLES FROM 0 TO 360
(5.3 and 5.4)
A. THE UNIT CIRCLE
The UNIT CIRCLE is a very compact and brilliant way of understanding trigonometric ratios,
special angles, and what is called the CAST RULE.
Imagine that we have a circle centred at the origin, with a radius of 1.
The line segment from the origin to point (1, 0) is called the initial arm. By rotating this arm
counter-clockwise, we can create a right-angled triangle against the x-axis. The hypotenuse, now
called the terminal arm, is length 1, and intersects the circle at a point P(x, y). Can we apply what
we already know about right-angle triangles here?
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These properties hold for any triangle we can inscribe in this unit circle, where the hypotenuse is the
radius (length 1). We always measure angles against the x-axis: thus, for any point P(x, y),
a) x = cos θ
b) y = sin θ
𝑠𝑖𝑛𝜃
c) 𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃
d) cos2 θ + sin2 θ = 1 (Pythag. Theorem)
Ex. For the given angles, use your calculators to determine the sine, cosine, and tangent of
each angle, to 3 decimal places. Draw each angle in standard position.
a)  = 50º
b)  = 115º
c)  = 220º
d)  = 330º
e)  = -95º
Any and all angles can be drawn in the Unit Circle by rotating the terminal arm the required number
of degrees. We can rotate an infinite number of times, in both directions, to create an infinite
number of angles.
POSITIVE ANGLES are formed when we rotate counter-clockwise.
NEGATIVE ANGLES are formed when we rotate clockwise.
CO-TERMINAL ANGLES are those which have the terminal arm located in the exact same
position. Co-terminal angles are formed by rotating 360 multiple times, either clockwise or counterclockwise. These angles share the exact same trig ratios because they are “located” in the exact same
position on the Unit Circle.
Ex. For each given angle: a)  = 45 and b)  = -120
(i) Draw the angle  in standard position.
(ii) Determine three other angles co-terminal with .
(iii) Write an expression to represent any angle co-terminal with .
Ex. Suppose P has rotated -790 about O (0, 0) from A(1, 0).
a) How many full rotations have been made?
b) In which quadrant is P located?
c) Draw a diagram to show the position of P.
These ratios apply even when the Unit Circle is dilated (magnified or shrunk). We simply use the
Unit Circle because of its ease in calculation.
Ex. A point P (3, 4) is located on a circle centred at the origin. Determine the 6 primary trig
ratios for the principal angle, and determine the angle to the nearest degree.
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B. SPECIAL ANGLES IN QUADRANT 1
Recall that we have two special triangles that give us exact measurements for their angles.
We can incorporate this knowledge into the Unit Circle (we’ll look closely at Quadrant I):
Remembering the “4-0” rule will help you visualize and remember the exact values for these ratios.
Ex. Without calculating, determine the 6 primary trig ratios of 30, 45 and 60 degrees.
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C. ALL ANGLES BETWEEN 0 AND 360 DEGREES
Special angles have an amazing property: they can be used to determine the exact trig ratios of
angles greater than 90 degrees.
Ex. Draw an angle of 150 degrees in the Unit Circle.
The 150 degree angle, which lies between the initial arm and the terminal arm, is called the principal
angle. The angle formed by the terminal arm and the x-axis, 30 degrees, is called the related acute angle
(notice that the two angles add up to 180 degrees; notice too that 150 is a multiple of 30).
The terminal arm intersects the unit circle in Quadrant II, at (-x, y). We can use the exact values for
30 degrees to determine these coordinates.
Let’s rotate even further, and create a principal angle of 210 degrees. What do you notice about this
angle?
Finally, let’s rotate once more and create a principal angle of 330 degrees. What do you notice here?
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This reveals to us an amazing property of angles and the Unit Circle.
Multiples of special angles (30, 45, 60) share the exact same trig ratios as these angles. The only
thing that changes, depending on which quadrant they lie in, is the sign of these values. In order to
remember these more easily, we use the CAST RULE.
S
A
T
C
In Quadrant I, ALL trig ratios are positive (A).
In Quadrant II, SINE ratios are positive (S) – the others are negative.
In Quadrant III, TAN ratios are positive (T) – the others are negative.
In Quadrant IV, COSINE ratios are positive (C) – the others are negative.
Ex. Determine the EXACT VALUES of the sin, cos, and tan of the following angles.
a)  = 135
b)  = -45
c)  = 225
d)  = 150
e)  = 240
f)  = -150
g)  = -300
Ex. Determine the EXACT VALUES for the following trig ratios.
a) csc120
b) sec(-315)
c) cot225
d) csc405
e) sec690
f) cot(-780)
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D. SOLVING FOR ANGLES BETWEEN 0 AND 360 DEGREES
Consider this question: “solve cos θ = ½, 0 < θ < 360”
We know that θ is a special angle by virtue of its exactness. By using our special triangle, we
know that θ = 60. But is there another, equally valid solution?
Consider the CAST rule and the Unit Circle. What can they tell us about this problem?
For ANY* unknown trig ratio, there are 2 equal and valid angle solutions between 0 and 360
degrees. Both answers must be given in order to fully solve the question. There are 3 handy
rules by which we can remember this:
Steps:
1) Use your calculator/special triangle to solve the trig ratio (call this angle θ1)
2) For COSINE: Your second angle θ2 = 360 - θ1
3) For SINE: θ2 = 180 - θ1
4) For TAN: θ2 = 180 + θ1
Question: what happens if your calculator gives you a negative angle?
Solution: use CO-TERMINAL ANGLES to assist you.
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Ex. Determine two angles between 0 and 360 that have each trigonometric value. Round
angles to the nearest degree.
a) sin = 0.672
b) cos = 0.291
c) tan = 1.573
d) sin = -0.875
Ex. Solve each equation for 0    360. Round each angle to the nearest degree.
a) sin = 0.39
b) cos = -0.64
c) tan = -2.15
Ex. Solve for both values of  between 0 and 360. Round to the nearest degree.
a) csc = 2.053
b) sec = 1.491
c) cot = 0.882
d) csc = -1.564
e) sec = -3.976
f) cot = -3.157
E. JUST A FEW MORE SPECIAL ANGLES
Let’s return to the Unit Circle for a few more “special angles”.
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