Here’s what you need to know about the Unit Circle:

1. A unit circle is a circle with a radius (line half circle) of 1 and because it has a radius of 1, you can measure sin, cos and tan with it.
2. The "sides" can be positive or negative according to the rules of Cartesian coordinates. This makes the sine, cosine and tangent change between positive and negative values also.
3. There are additional ratios for you to remember: cosec or csc A = hypotenuse / opposite, sec A = hypotenuse / adjacent and cotan/cot/cotg/ctg/ctn A = adjacent / opposite
4. There is another unit of measurement and that is called radians (rad / r) and you can use that measurement to show the circumference of a circle (2𝜋r).
   
5. Degree to radians = divide by 180 then times by 𝜋. Radians to degrees is the reverse, divide by 𝜋 and then multiply by 180.
6. The table showing when sin, cos and tan are positive or negative based on the unit circle.

By Jim.belk - Own work, CC BY-SA 3.0, Link

Example Question/s

Watch video for explanation of the following question/s:

Question 1

sin(30°)-cos(60°)= ?
a) tan(0°)
b) sin(0°)
c) -30°
d) 0
This question asks us to find the value of sin(30°) and cos(60°), and then subtract one from the other.

We know that 30° and 60°are both 30°away from the borders of their quadrant (in other words, 0 + 30° = 30° and 90°- 60°= 30°), and so their coordinates on the unit circle are the inverse of each other. That means that if the coordinates of 30° are (a, b), then the coordinates of 60°are (b, a).

We also know that the x-coordinate of a point on the unit circle is cos(a), and the y-coordinate of a point on the unit circle is sin(a). Therefore, if the coordinates of 30° are (cos(30°), sin(30°)), the coordinates of 60° are (sin(30°), cos(30°) OR (cos(60°), sin(60°). This means that cos(60°) is equivalent to sin(30°).

Therefore, we can simplify our equation to sin(30°)-sin(30°), which of course must equal zero (since any number minus itself equals zero). Our answer is (d). To recap:

1. Find the position of the two angles on the unit circle
2. Find their coordinates
3. Solve the equation for the missing value

Question 2

What is the supplementary angle to the angle 6𝜋/8?
a) 42°
b) 2𝜋/8
c) -6𝜋/8
d) 4°

This question asks us to understand angles in radians, such as 6𝜋/8. If it makes you more comfortable, you can convert this angle to degrees by multiplying it by 180/𝜋 to get a value of 135°- however, you can also solve the question entirely in radians.

We know that supplementary angles sum to 180° or 𝜋 radians, so we need to find an angle which, when added to 6𝜋/8, equals 𝜋 radians. We can express this in an equation as 𝜋 =6𝜋/8 + ?. In order to put all values on the same base (8), we can rewrite this as 8𝜋/8=6𝜋/8+?𝜋/8- meaning that ?+6=8.

This means that our unknown value is 2, and therefore the angle is 2𝜋/8. Our answer is (b). To recap:

1. If required, convert all angles in radians to degrees by multiplying by 180/𝜋
2. Express the sum of the angles as an equation equalling 180°or 𝜋
3. Solve for the unknown angle

Question 3

If sin(a°) = -1/√3 and 0 ≤ a ≤ 3𝜋/2, what quadrant is a° in?
a) Quadrant 1
b) Quadrant 2
c) Quadrant 3
d) Quadrant 4

We know that sin(a°)= -1/√3,and we want to know which quadrant of the unit circle the angle a° is in. We know that the domain of a is between 0 and 3𝜋/2, meaning that it must be in the first quadrant (0 - 𝜋/-2), second quadrant (𝜋/2 - 𝜋), or third quadrant (𝜋, 3𝜋/2). Therefore, we can knock out option (d).

Next, we need to think about which quadrants sin is positive in. The following table summarises which values are positive and negative in each quadrant.

From this table, we can see that if sin(a°) is negative, “a” must be in the 3rd or 4th quadrant - but since we have already decided that it can’t be in the 4th quadrant, the answer must be (c). To recap:

1. Look carefully at the excluded values of a°
2. Work out which quadrants sin is negative in

Question 4
Which of these points is on the unit circle?
a) (1, 1)
b) (√(3)/2, 1/2)
c) (14/3, 2)
d) (1/4, 3/4)

To answer this question, we need to use the formula of the unit circle to work out which point is on this line.
The formula of the unit circle is the standard formula of a circle with radius 1 and center (0, 0) - y² + x² = 1. We need to substitute the points in each of these options into that equation to work out if the statement is correct.

(1)² + (1)² = 1, 1 + 1 = 1, ∴ Incorrect
(√(3)/2)² + (1/2)² = 1, 3/4+ 1/4= 1, ∴ Correct
(14/3)² + (2)² = 1, 196/9+ 4 = 1, ∴ Incorrect
(1/4)² + (3/4)² = 1, 1/16+9/16= 1, ∴ Incorrect

The only one of these points on the unit circle is (√(3)/2, 1/2), since it’s the only one which satisfies the equation y² + x² = 1 and this means the answer is (b). To recap:

1. Recall the equation of the unit circle
2. Work out which coordinates fit the equation

Key Rules to remember

• A unit circle is a circle with a radius (line half circle) of 1 and because it has a radius of 1, you can measure sin, cos and tan with it.
• The "sides" can be positive or negative according to the rules of Cartesian coordinates. This makes the sine, cosine and tangent change between positive and negative values also.
• There are additional ratios for you to remember: cosec or csc A = hypotenuse / opposite, sec A = hypotenuse / adjacent and cotan/cot/cotg/ctg/ctn A = adjacent / opposite
• There is another unit of measurement and that is called radians (rad / r) and you can use that measurement to show the circumference of a circle (2𝜋r).
  
• Degree to radians = divide by 180 then times by 𝜋. Radians to degrees is the reverse, divide by 𝜋 and then multiply by 180.
• Knowing the values in the table showing when sin, cos and tan are positive or negative based on the unit circle may be useful for your exam.

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