Finding Coordinates at 0 Degrees

What are the coordinates of the point on the unit circle at $\theta = 0°$?

Locate the angle: Start at the positive x-axis and rotate $0°$ (no rotation) = Point is at the rightmost position

Find the x-coordinate (cosine): The point is 1 unit to the right of the center = $\cos(0°) = 1$

Find the y-coordinate (sine): The point is 0 units above or below the center = $\sin(0°) = 0$

Answer: The coordinates are $(1, 0)$, so $\cos(0°) = 1$ and $\sin(0°) = 0$.

Finding Coordinates at 90 Degrees

What are the coordinates of the point on the unit circle at $\theta = 90°$?

Locate the angle: Start at positive x-axis and rotate $90°$ counterclockwise = Point is at the top of the circle

Find the x-coordinate (cosine): The point is directly above the center, so $x = 0$ = $\cos(90°) = 0$

Find the y-coordinate (sine): The point is 1 unit above the center = $\sin(90°) = 1$

Answer: The coordinates are $(0, 1)$, so $\cos(90°) = 0$ and $\sin(90°) = 1$.

Understanding the 45-Degree Angle

Find $\sin(45°)$ and $\cos(45°)$ using the unit circle.

Locate the angle: At $45°$, the point is exactly halfway between $0°$ and $90°$ = The x and y coordinates are equal

Use the unit circle equation: Since $x = y$ and $x^2 + y^2 = 1$, we get $2x^2 = 1$ = $x^2 = \frac{1}{2}$

Solve for x and y: $x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ = $\sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2}$

Answer: At $45°$, both $\sin(45°)$ and $\cos(45°)$ equal $\frac{\sqrt{2}}{2} \approx 0.707$.

Finding Values in the Second Quadrant

What are $\sin(120°)$ and $\cos(120°)$?

Find the reference angle: $180° - 120° = 60°$ = Reference angle is $60°$

Determine the quadrant: $120°$ is between $90°$ and $180°$ = Quadrant II: x is negative, y is positive

Apply signs to the values: Use $60°$ values: $\sin(60°) = \frac{\sqrt{3}}{2}$, $\cos(60°) = \frac{1}{2}$ = $\sin(120°) = \frac{\sqrt{3}}{2}$, $\cos(120°) = -\frac{1}{2}$

Answer: $\sin(120°) = \frac{\sqrt{3}}{2}$ and $\cos(120°) = -\frac{1}{2}$.

Mistake: Confusing which coordinate is sine and which is cosine

Why: It's easy to mix up x and y when both relate to the angle.

Correct: Remember: **Cosine = x-coordinate** (horizontal), **Sine = y-coordinate** (vertical). Think 'x comes before y alphabetically, and cosine comes before sine.'

Mistake: Forgetting that angles are measured counterclockwise

Why: By convention, positive angles go counterclockwise from the positive x-axis.

Correct: Always start at the positive x-axis (3 o'clock position) and rotate counterclockwise for positive angles.

Mistake: Using wrong signs in different quadrants

Why: Students often forget that coordinates can be negative.

Correct: Use the ASTC rule: **A**ll (Q1), **S**ine (Q2), **T**angent (Q3), **C**osine (Q4) - tells you which functions are positive.

Ferris Wheel Motion

A Ferris wheel rotates in a circle. Your height above ground follows a sine wave pattern as you ride!

On a Ferris wheel with radius 10 meters, your height varies between 0 and 20 meters as you complete each rotation.

Clock Hands

Clock hands rotate around the center, and their position can be described using the unit circle.

At 3:00, the minute hand points at 12 (top), which is the 90-degree position on the unit circle.

Q: Why is it called the 'unit' circle?

A: It's called the unit circle because its radius is exactly 1 unit. This makes calculations simpler because multiplying by 1 doesn't change values.

Q: How do I remember whether sine or cosine is x or y?

A: Think alphabetically: 'c' comes before 's', and 'x' comes before 'y'. So **c**osine goes with **x**, and **s**ine goes with **y**.

Q: What is the difference between degrees and radians?

A: Both measure angles. Degrees divide a circle into 360 parts. Radians use the circle's radius: a full circle is $2\pi$ radians. To convert: $180° = \pi$ radians.