Trigonometry Unit Circle Overview

Absolutely, Ella! Here’s a full script transcript for a video lecture that teaches the entire topic in an engaging, structured, and student-friendly tone. You can read it like a voiceover or use it as a script for a presentation.

[Opening Slide – Title: “Mastering Trigonometry with the Unit Circle”]

[Speaker begins]

Hi everyone! I’m Ella Pollak, and welcome to today’s lecture on Mastering Trigonometry with the Unit Circle. By the end of this session, you’ll have a solid grasp of the unit circle, special triangles, trig functions, and how they all come together — including inverse and reciprocal functions. Let’s jump in.

[Slide: The Unit Circle & Quadrants]

So, what is the unit circle?

It’s a circle centered at the origin, with a radius of exactly 1 unit. It’s like the home base for trigonometry, and it’s divided into four quadrants, going counterclockwise from the positive x-axis.

We remember which trig functions are positive in each quadrant using this phrase:

👉 “All Students Take Calculus.”

• All — All functions positive in Quadrant I
• Students — Sine only in Quadrant II
• Take — Tangent only in Quadrant III
• Calculus — Cosine only in Quadrant IV

Also remember:

• Cosine is the x-coordinate
• Sine is the y-coordinate

Let’s build on this with angles.

[Slide: Important Angles & Coordinates]

There are some angles you must memorize — in both degrees and radians. These are:

• 0°, 30°, 45°, 60°, 90°, 120°, all the way up to 360°
• In radians, that’s 0, π/6, π/4, π/3, π/2, etc., up to 2π

Let’s look at the coordinates for key angles in Quadrant I:

• 0° → (1, 0)
• 30° → (√3/2, 1/2)
• 45° → (√2/2, √2/2)
• 60° → (1/2, √3/2)
• 90° → (0, 1)

These values repeat in other quadrants — the coordinates stay the same, but the signs change depending on where you are on the circle.

[Slide: Reference Angles & Signs]

Next up: Reference angles.

A reference angle is the acute angle between your terminal side and the x-axis. It helps us evaluate trig functions even for angles outside Quadrant I.

Here’s the key:

Find the reference angle → use known trig values → adjust the sign based on the quadrant.

Example:

tan(135°) → Reference angle is 45°

→ tan(45°) = 1

→ But 135° is in QII, where tangent is negative ⇒ tan(135°) = –1

[Slide: Special Triangles]

Let’s talk about special triangles. These are life-savers in trig.

1. 30°-60°-90° triangle:

• Side ratios: 1, √3, 2
• Opposite 30°: 1
• Opposite 60°: √3
• Hypotenuse: 2

2. 45°-45°-90° triangle:

• Side ratios: 1, 1, √2
• Works great for angles like 45°, 135°, 225°, and 315°

Use SOH-CAH-TOA to relate the sides to the angles:

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent

[Slide: Evaluating Trig Functions Without a Unit Circle]

If you don’t have the unit circle in front of you, no worries. You can still work things out.

1. Find the reference angle
2. Use a special triangle to get the side ratios
3. Adjust for the quadrant’s sign
4. And finally — rationalize your answer if needed

Example:

cos(150°) → reference angle is 30°

→ cos(30°) = √3/2

→ In QII, cosine is negative ⇒ cos(150°) = –√3/2

[Slide: Reciprocal Trig Functions]

Next up — reciprocal trig functions.

These are just the flipped versions of sine, cosine, and tangent:

• sec(θ) = 1 / cos(θ)
• csc(θ) = 1 / sin(θ)
• cot(θ) = 1 / tan(θ)

Important: these functions are undefined when the denominator is 0.

Example:

If sin(90°) = 1, then csc(90°) = 1/1 = 1

But sec(90°)? Undefined — because cos(90°) = 0!

[Slide: Finding All Trig Functions from One]

Sometimes you’re given just one trig function and a quadrant, and asked to find the rest.

Steps:

1. Sketch a triangle
2. Use Pythagoras to find the missing side
3. Apply SOH-CAH-TOA
4. Check signs by quadrant

Example:

Given: sin(θ) = 3/5, and θ is in QII

→ Opposite = 3, Hypotenuse = 5

→ Adjacent = √(5² – 3²) = 4 → so x = –4 (since QII)

Now:

• sin(θ) = 3/5
• cos(θ) = –4/5
• tan(θ) = 3/–4 = –3/4
  …and from there, you can get the reciprocals.

[Slide: Inverse Trig Functions & Restrictions]

Inverse trig functions tell you the angle when you know the trig ratio.

But here’s the catch: they’re restricted to specific quadrants:

• sin⁻¹(x) and tan⁻¹(x) give answers between –90° and 90° → Quadrants I & IV
• cos⁻¹(x) gives answers between 0° and 180° → Quadrants I & II

Example:

• sin⁻¹(1/2) = 30° (NOT 150°, even though both have sin = 1/2)
• cos⁻¹(–√2/2) = 135°, not 225°

Always check if your answer fits in the right range.

[Slide: Key Definitions Recap]

Let’s recap some key terms:

• Unit Circle: Circle with radius 1 centered at origin
• Reference Angle: Acute angle to the x-axis
• SOH-CAH-TOA: Trig ratio mnemonic
• Reciprocal Trig Functions: sec, csc, cot
• Inverse Trig Functions: Give angles from a trig ratio, with limited range

[Slide: Action Items – What to Study Next]

Now it’s your turn. Here’s what to focus on:

✅ Memorize all unit circle angles, degrees & radians

✅ Practice sketching triangles and using SOH-CAH-TOA

✅ Review special triangles and reciprocal identities

✅ Work through practice problems on all the above

[Closing Slide – You’ve Got This!]

That’s it for this video — you’ve covered a huge chunk of trig.

Keep reviewing and practicing, and you’ll smash that test.

Thanks for watching, and see you next time!