Trigonometry is the maths of angles. Before we can do anything clever with sine, cosine, and tangent — the three functions at the heart of the subject — we need a clear, shared way to talk about angles themselves. That is what this first lesson is for.
We will cover four things, each building on the last: how to place an angle on the coordinate plane, why many different-looking angles are secretly the same, how to tell which "corner" of the plane an angle points into, and the two units we measure angles in. None of it is hard, but it is the foundation everything else rests on — so we will take it slowly and make sure each piece clicks.
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Placing an angle: standard position
The simplest way to think about an angle is as a turn. You start facing one direction, rotate, and stop facing another. The size of the angle is just how far you turned.
So that everyone draws and measures angles the same way, we always begin them from the same place. An angle is in standard position when two things are true: its corner — the vertex — sits at the origin, and its starting arm — the initial side — lies along the positive x-axis. From there we rotate until we reach the terminal side, the arm we finish on.
An angle α in standard position: vertex at the origin, initial side on the positive x-axis, terminal side swept counter-clockwise.
Direction matters, and the rule is the opposite of how a clock turns: we count counter-clockwise as positive and clockwise as negative. So +90° is a quarter-turn upward, finishing on the positive y-axis, while −90° is a quarter-turn the other way, finishing on the negative y-axis. Same amount of turning, opposite directions.
Standard position. Vertex at the origin; initial side along the positive x-axis. Counter-clockwise rotation makes a positive angle; clockwise makes a negative angle. The terminal side is the arm you finish on.
An angle can keep turning past a full circle. 360° is one complete turn, right back to the start. 450° is a full turn plus another 90°, so it finishes pointing the same way as 90° — even though it travelled further to get there.
Try it! · Check-in question
In standard position, the terminal side of a 90° angle points along which axis?
Start on the positive x-axis and turn counter-clockwise by 90° (a quarter-turn). That lands you on the positive y-axis, so the answer is B. Option A is where 0° points — the initial side, before any turning. Option C (negative x-axis) is 180°, a half-turn. Option D (negative y-axis) is 270°, or equivalently −90°.
Co-terminal angles: same finish, different journey
Here is the first idea that catches people out. Because a full turn of 360° brings you right back to where you started, you can add or subtract a whole turn and end up pointing in exactly the same direction. The two angles are written differently, but they share the same terminal side. We call them co-terminal.
Picture walking around a circular running track. Whether you walk a quarter of the way around, or a quarter plus one full lap, or a quarter minus a lap (one lap backwards), you finish standing in the same spot. The extra laps change how far you walked, but not where you end up.
The terminal side reached by 40° is the very same arm reached by 400° (40° + one turn) and −320° (40° − one turn). Co-terminal angles differ only by whole turns.
Finding a co-terminal angle. Add or subtract whole turns — 360° in degrees, or 2π in radians — as many times as you like. Each whole turn you add or remove lands you on the same terminal side again.
Worked example · finding co-terminal angles
Find three angles that are co-terminal with 40°.
Add one full turn: 40° + 360° = 400°.
Add another full turn: 400° + 360° = 760°.
Or subtract a full turn instead: 40° − 360° = −320°.
So 400°, 760°, and −320° all share the same terminal side as 40°. There are infinitely many — one for every number of laps, forwards or backwards.
Co-terminal angles. Two angles are co-terminal when they share the same terminal side. They differ by a whole number of full turns: a multiple of 360° (or of 2π radians).
Try it! · Check-in question
Which of the following angles is co-terminal with 30°?
Co-terminal angles differ by a whole number of full turns (360°). 390° − 30° = 360°, exactly one turn, so 390° works. The answer is D. Option A (210°) adds half a turn (30° + 180°), so it points the opposite way. Option B (−30°) just changes the sign; that turns 30° the other way and lands below the x-axis. Option C (330°) subtracts the angle from a full turn (360° − 30°) instead of adding a full turn; 330° is co-terminal with −30°, not with 30°.
The four quadrants
The x-axis and the y-axis cross at the origin and slice the plane into four regions called quadrants. We number them counter-clockwise, starting from the top-right: Quadrant 1 is top-right, Quadrant 2 top-left, Quadrant 3 bottom-left, and Quadrant 4 bottom-right.
The four quadrants, numbered counter-clockwise from the top-right — with the angle range each one covers and the sign pattern (x, y) of points inside it.
Two kinds of exam question ask about quadrants, and they turn out to be the same idea seen from two sides. One asks which quadrant a point sits in. The other asks which quadrant an angle's terminal side points into. Let us take them one at a time.
Which quadrant is a point in?
This is decided entirely by the signs of the two coordinates — nothing else. If x is positive the point is on the right; if x is negative it is on the left. If y is positive it is up; if y is negative it is down. Put those together and you land in one quadrant.
In the table below, green marks a positive coordinate and red marks a negative one.
Quadrant
x-coordinate
y-coordinate
Q1 — top-right
positive (+)
positive (+)
Q2 — top-left
negative (−)
positive (+)
Q3 — bottom-left
negative (−)
negative (−)
Q4 — bottom-right
positive (+)
negative (−)
Worked example · quadrant of a point
In which quadrant is the point (−3, −5)?
Look only at the signs.
The x-coordinate, −3, is negative (so: left).
The y-coordinate, −5, is negative (so: down).
Left and down is the bottom-left corner — Quadrant 3.
Sometimes the signs are hidden behind a little reasoning. For example: "if a < 0 and ab < 0, where is the point (a, b)?"
Since a is negative and the product ab is also negative, b must be positive (negative × positive = negative).
So (a, b) has signs (−, +): that is Quadrant 2.
Try it! · Check-in question
Which of the following points lies in the third quadrant?
Quadrant 3 is the bottom-left, where both coordinates are negative. Only (−2, −3) fits, so the answer is A. Option B (2, 3) is (+, +), Quadrant 1. Option C (−2, 3) is (−, +), Quadrant 2. Option D (2, −3) is (+, −), Quadrant 4.
Which quadrant does an angle point into?
For an angle that is already between 0° and 360°, just see which range it falls in: 0°–90° is Q1, 90°–180° is Q2, 180°–270° is Q3, and 270°–360° is Q4. (In radians the same ranges are 0 to π/2, π/2 to π, π to 3π/2, and 3π/2 to 2π.)
For a negative angle, or one bigger than a full turn, you cannot read it off directly — first replace it with a co-terminal angle in the 0°–360° range, using the add-or-subtract-a-turn trick from earlier, and then read the quadrant.
1
If the angle is negative, or 360° (2π) or larger, add or subtract whole turns until it lands between 0° and 360°.
2
Read off the quadrant from where it falls: 0°–90° → Q1, 90°–180° → Q2, 180°–270° → Q3, 270°–360° → Q4.
Worked example · quadrant of a large negative angle
In which quadrant does the terminal side of −498° lie?
It is negative, so add a full turn to climb toward the 0°–360° range: −498° + 360° = −138°.
Still negative, so add another: −138° + 360° = 222°.
Now 222° is in range, and it sits between 180° and 270° — so the terminal side is in the third quadrant.
Common mistake. A negative angle does not automatically point "downward" into Q3 or Q4. For instance, −30° does land in Q4, but −200° lands in Q2. The minus sign alone tells you nothing — always bring the angle into the 0°–360° range first, then read the quadrant.
Try it! · Check-in question
In which quadrant does the terminal side of 480° lie?
480° is bigger than a full turn, so subtract one: 480° − 360° = 120°. Since 120° lies between 90° and 180°, the terminal side is in the second quadrant (B). Option A would need a co-terminal angle between 0° and 90°. Options C and D cover 180°–270° and 270°–360°, and 120° is in neither. The usual slip is trying to place 480° without first subtracting the full turn.
Two ways to measure: degrees and radians
So far we have measured in degrees, where a full circle is 360°. Degrees are familiar and easy to picture, but the number 360 is just a historical choice — the Babylonians liked it, partly because a year is roughly 360 days. There is nothing mathematically special about it.
Radians are the other unit, and they come from the circle itself rather than from tradition. One radian is the angle you get when the arc along the circle is exactly as long as the radius. Wrap the radius along the edge of the circle once, and the angle it spans at the centre is one radian.
One radian is the angle cut off by an arc as long as the radius. The whole circumference is 2πr — about 6.28 radius-lengths — so a full turn is 2π radians.
Because the circumference of a circle is 2πr — that is, 2π radius-lengths laid end to end — a full turn is exactly 2π radians. Half of that, a half-turn, is π radians. But a half-turn is also 180°. So the two units meet at one tidy fact:
The bridge: 180° = π radians. Everything about converting between the units comes from this single equation. From it, degrees → radians: multiply by π/180. Radians → degrees: multiply by 180/π.
Why go to the trouble of radians? Because they make formulas clean. For example, an arc that subtends an angle θ (in radians) on a circle of radius r has length s = rθ — no stray conversion factor. That neatness is why higher mathematics, and the CSCA exam, lean on radians. You do not need to memorise s = rθ for this lesson; it is just the reason radians are worth knowing.
A good way to remember which factor to use: you are swapping the unit you have for the one you want, so multiply by the fraction that cancels the unit you are leaving. Leaving degrees? Multiply by π/180 (degrees on the bottom, so they cancel). Leaving radians (which carry the π)? Multiply by 180/π.
Mental shortcut. Since π/6 = 30°, any multiple of π/6 is just that multiple of 30°: π/6 = 30°, π/3 = 60°, π/2 = 90°, and so on. Likewise π/4 = 45°, so any multiple of π/4 is that multiple of 45°. With these two facts you can convert most common angles in your head.
Degrees
Radians
0°
0
30°
π/6
45°
π/4
60°
π/3
90°
π/2
180°
π
270°
3π/2
360°
2π
Watch the direction. Multiplying by 180/π when you should multiply by π/180 (or the reverse) is the most common slip here, and it produces a wildly wrong answer. Quick sanity check: 90° must turn into π/2, and π/2 must turn back into 90°. If your result fails that check, you used the upside-down factor.
Worked example · degrees to radians
Convert 135° to radians.
Leaving degrees, so multiply by π/180: 135° × π/180 = 135π/180.
Now simplify the fraction 135/180.
Both top and bottom divide by 45: 135 ÷ 45 = 3 and 180 ÷ 45 = 4.
So 135π/180 = 3π/4.
Try it! · Check-in question
Convert 150° to radians.
Multiply by π/180: 150 × π/180 = 150π/180. Simplify 150/180 by dividing both by 30: that gives 5/6, so the answer is 5π/6 (C). Option A (6π/5) turns the fraction upside down: 180/150 = 6/5. Option B (5π/12) uses 360 in place of 180: 150/360 = 5/12. Option D (5/6) simplifies correctly but drops the π, and 5/6 of a radian is a much smaller angle.
Worked example · radians to degrees
Convert 7π/6 to degrees.
Leaving radians, so multiply by 180/π: (7π/6) × (180/π).
The π on top cancels the π on the bottom, leaving 7 × 180/6 = 7 × 30 = 210°.
Spotting that the denominator (6) divides neatly into 180 keeps the arithmetic to a single multiplication.
Try it! · Check-in question
Express −8π/3 as an angle in degrees.
Multiply by 180/π: (−8π/3) × (180/π). The π cancels, leaving −8 × 180/3 = −8 × 60 = −480°. The answer is A. Option B (−960°) uses 360/π, which doubles everything. Option C (−120°) first adds a full turn (−8π/3 + 2π = −2π/3) and converts that. −120° is co-terminal with −480°, but it is a different angle measure, and the question asks for −8π/3 itself. Option D (−60°) forgets the 8 and converts only π/3 = 60°. (As a bonus, −480° is co-terminal with 240°, which sits in Q3.)
Your turn! · Practice questions
Practice · Question 1
Which angle, in radians, is co-terminal with −45°?
Two steps. Convert: −45° × π/180 = −π/4. Then add one full turn, 2π = 8π/4: −π/4 + 8π/4 = 7π/4. (In degrees: −45° + 360° = 315°, and 315° = 7π/4.) The answer is D. Option A (π/4) drops the minus sign; π/4 is above the x-axis, but −π/4 is below it. Option B (3π/4) adds half a turn (π) instead of a full turn. Option C (9π/4) drops the minus sign and then adds a turn: π/4 + 2π = 9π/4, which is co-terminal with +45°.
Practice · Question 2
About how many degrees is an angle of 1 radian? (Use π ≈ 3.14.)
Leaving radians, so multiply by 180/π: 1 × 180/π ≈ 180/3.14 ≈ 57.3°. The answer is C. This fits the picture of a radian: the arc is as long as the radius, a little less than one-sixth of a full turn (60°). Option A (0.017°) multiplies by π/180, the upside-down factor. Option B (114.6°) uses 360/π, which doubles the answer. Option D (180°) mixes up 1 radian with π radians; it is π radians that equal 180°.
Practice · Question 3
If a > 0 and b < 0, in which quadrant is the point (b, −a)?
Work out the sign of each coordinate. The x-coordinate is b, and b < 0, so x is negative. The y-coordinate is −a; since a is positive, −a is negative. Signs (−, −) mean the bottom-left: the third quadrant, so the answer is C. Option B (Q2) ignores the minus in −a and reads y as positive. Option D (Q4) reads the point as (a, b) = (+, −), swapping the coordinates. Option A (Q1) would need both coordinates positive, which only happens if both signs are misread.
Practice · Question 4
In which quadrant does the terminal side of −5π/3 lie?
The angle is negative, so add one full turn, 2π = 6π/3: −5π/3 + 6π/3 = π/3. That is 60°, between 0° and 90°, so the terminal side is in the first quadrant (A). Option D (Q4) is the trap from the "Common mistake" box: thinking a negative angle must point downward. Dropping the minus sign gives the same wrong answer, because 5π/3 = 300° is in Q4. Option C (Q3) adds half a turn (π) instead of a full turn: −5π/3 + π = −2π/3 = −120°, which is in Q3. Option B (Q2) would need an angle between 90° and 180° after the turn is added, and π/3 is not one.
Practice · Question 5
Which of the following angles is co-terminal with 7π/3?
Subtract one full turn, which is 2π = 6π/3: 7π/3 − 6π/3 = π/3. So 7π/3 is co-terminal with π/3, and the answer is B. Option A (4π/3) subtracts half a turn (π = 3π/3) instead of a full turn. Option C (−π/3) subtracts the wrong way round: 2π − 7π/3 = −π/3. Option D (5π/3) writes one full turn as 2π/3 instead of 6π/3, so it does 7π/3 − 2π/3.
Things to remember
Concept
What to know
Standard position
Vertex at the origin, initial side on the +x axis. Counter-clockwise is positive, clockwise is negative.
Co-terminal
Same terminal side. Angles that differ by a whole number of full turns (360° or 2π) are co-terminal.
Quadrant of a point
From the signs: Q1 (+, +), Q2 (−, +), Q3 (−, −), Q4 (+, −).
Quadrant of an angle
0°–90° → Q1, 90°–180° → Q2, 180°–270° → Q3, 270°–360° → Q4. Reduce into 0°–360° first if needed.
The bridge
180° = π radians. A full turn is 360° = 2π.
Converting
Degrees → radians: × π/180. Radians → degrees: × 180/π. (Multiply by the fraction that cancels the unit you're leaving.)
