Tri

1. Trigonometric functions lecture 2

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TRIGONOMETRIC
FUNCTIONS
LECTURE 2
Abdullayev Azizbek
10/6/2026
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2. Measuring Angles — From Zero to 360° (and Beyond!)

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MEASURING ANGLES — FROM ZERO TO
360° (AND BEYOND!)
Let's imagine that an angle is not just two lines meeting at a point. An angle is a rotation. Think of a clock hand, a
spinning pizza, or you turning around because someone called your name in class. Once you understand rotation,
measuring angles becomes much easier.
Imagine you're standing in the middle of a classroom.
You face forward. That's your initial side.
You start turning.
Wherever you stop, that's your terminal side.
The amount you turn is your angle.
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3. Angles

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ANGLES
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5. Why is a full rotation 360°?

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WHY IS A FULL ROTATION 360°?
Imagine a pizza cut into 360 tiny equal slices. If you eat all 360 slices, you've eaten the entire pizza. (Please don't try
this). Mathematicians divide one complete revolution into 360 equal parts, and each part is called one degree.
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6.

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8. What are degrees, minutes, and seconds?

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WHAT ARE DEGREES, MINUTES, AND
SECONDS?
A degree can be divided into smaller units, just as an hour can be divided into minutes and seconds.
1∘ = 60′
1′ = 60′′
Therefore,
1∘ = 3600′′
∘ means degrees.
′ means minutes (also called arcminutes).
′′ means seconds (also called arcseconds).
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12.

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13.

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14.

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15.

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16.

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17. Test your understanding

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TEST YOUR UNDERSTANDING
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18.

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19.

Radians
Angles, but make it delicious
No more "why 2π?!" faces. Promise.

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Why do radians even exist?
Because 360 is a weird number
Degrees
Radians
Ancient Babylonians loved the
number 60, so we got 360.
Made from the circle itself: the
radius is the ruler.
It is basically a random choice.
No random numbers involved.
Like measuring your pizza in "Kevin's
thumbs".
Makes sin, cos, and calculus much
cleaner later.

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What is 1 radian?
The pizza-string trick
1
Cut a string as long as the radius
(center to crust).
2
Lay that string along the crust.
3
The angle it covers from the center is
1 radian.
r
1 rad
r (crust)
1 radian ≈ 57.3° (just under 1 slice of a 6-slice
pizza)

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Conversion magic
Two moves, that's it
Degrees → Radians
Radians → Degrees
multiply by π/180
multiply by 180/π
π on top!
180 on top!
Memory trick: put on top the unit you WANT. Want radians? π on top. Want degrees? 180 on
top. The old unit cancels out. Magic

24.

Degrees → radians: examples
Multiply by π/180, then simplify
60°
135°
270°
Step 1
Step 1
Step 1
60 × π/180 = 60π/180
135 × π/180 = 135π/180
270 × π/180 = 270π/180
Step 2: simplify
Step 2: simplify
Step 2: simplify
÷60 → π/3
÷45 → 3π/4
÷90 → 3π/2

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Radians → degrees: examples
Multiply by 180/π (π cancels!)
π/6
5π/4
2 rad
(π/6) × (180/π) = 180/6
(5π/4) × (180/π) = 900/4
2 × 180/π = 360/π
= 30°
= 225°
≈ 114.6°
Shortcut: if it has π in it, just swap π for 180 and simplify. π/6 → 180/6 = 30°

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The cheat sheet (screenshot me!)
Memorize the first three; the rest are multiples
0°
30°
45°
60°
90°
120°
135°
150°
180°
270°
360°
0
π/6
π/4
π/3
π/2
2π/3
3π/4
5π/6
π
3π/2
2π
Pattern: 30° = π/6, 45° = π/4, 60° = π/3.
Double 30° → 60° → π/3. Double 45° → 90° → π/2. Everything else is just adding slices!

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29. Trigonometric Functions

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TRIGONOMETRIC FUNCTIONS
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30. Reciprocal Trigonometric Functions

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RECIPROCAL TRIGONOMETRIC
FUNCTIONS
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35.

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We can also represent the trigonometric functions on a cartesian plane in terms of