Sine and Cosine Rules, Angle Measures, the Unit Circle and Vectors (Extended)

Mathematics · IB Middle Years Programme (MYP) · 9 min read · Updated: 22 Oct 2026

Welcome to the World of Trigonometry!

Hi there! Today we are diving into one of the most visual and useful chapters in Mathematics: Trigonometric Functions and the Unit Circle. Don't let the long name scare you! If you have ever played a video game with moving characters, watched the tide come in and out, or even just used a GPS, you have seen trigonometry in action.

In this chapter, we are going to move beyond simple right-angled triangles and learn how circles and waves help us understand the world around us. Let’s get started!

1. Beyond Degrees: Meeting the Radian

Up until now, you have probably measured angles in degrees (\(0^\circ\) to \(360^\circ\)). But in MYP Year 5, we introduce a new "language" for angles: Radians.

What is a Radian? Imagine taking the radius of a circle and wrapping it around the edge (the circumference). The angle created at the center is exactly 1 radian. Because a full circle has a circumference of \(2\pi r\), there are exactly \(2\pi\) radians in a full circle.

How to Convert Between the Two

Don't worry if this feels like learning a new currency; the exchange rate is simple!

To go from Degrees to Radians: Multiply by \( \frac{\pi}{180} \)
To go from Radians to Degrees: Multiply by \( \frac{180}{\pi} \)

Quick Review:
\(180^\circ = \pi\) radians
\(360^\circ = 2\pi\) radians
\(90^\circ = \frac{\pi}{2}\) radians

Key Takeaway: Radians are just another way to measure how far you have turned. Think of Degrees as "Fahrenheit" and Radians as "Celsius"—they measure the same thing, just on a different scale!

2. The Unit Circle: Your New Best Friend

The Unit Circle is a circle with a radius of exactly 1 unit, centered at the point \((0, 0)\). This circle is like a "cheat sheet" for all trigonometry.

When we pick a point \((x, y)\) anywhere on this circle:

The x-coordinate is always the Cosine of the angle: \( x = \cos(\theta) \)
The y-coordinate is always the Sine of the angle: \( y = \sin(\theta) \)
The Tangent is the slope, or: \( \tan(\theta) = \frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)} \)

Did you know? Because the radius is 1, we can use the Pythagorean theorem (\(a^2 + b^2 = c^2\)) to find a famous identity:
\( \sin^2(\theta) + \cos^2(\theta) = 1 \). This is true for any angle!

3. Which Way is Positive? (The CAST Diagram)

As you move around the circle, the coordinates (x and y) change from positive to negative. We divide the circle into four Quadrants. To remember where Sine, Cosine, and Tangent are positive, use the CAST rule (starting from the bottom-right and moving counter-clockwise) or ASTC (starting top-right):

Quadrant 1 (Top Right): All are positive (\(\sin, \cos, \tan\)).
Quadrant 2 (Top Left): Sine is positive (others are negative).
Quadrant 3 (Bottom Left): Tangent is positive (others are negative).
Quadrant 4 (Bottom Right): Cosine is positive (others are negative).

Memory Aid: Just remember the phrase "All Stations To City" or "Add Sugar To Coffee" to keep track of the quadrants!

4. Exact Values: No Calculator Needed!

In many IB exams, you will be asked for "exact values." This means they don't want a long decimal from a calculator; they want fractions and square roots. The most important angles to memorize are \(30^\circ\), \(45^\circ\), and \(60^\circ\).

The "Finger Trick" for Sine and Cosine:
Hold up your left hand, palm facing you. Imagine your pinky is \(0^\circ\), ring finger is \(30^\circ\), middle is \(45^\circ\), index is \(60^\circ\), and thumb is \(90^\circ\). To find the value for an angle, fold down that finger.
\( \sin = \frac{\sqrt{\text{fingers below}}}{2} \)
\( \cos = \frac{\sqrt{\text{fingers above}}}{2} \)

Key Takeaway: Practice these values! Being able to recognize that \(\sin(30^\circ) = \frac{1}{2}\) or \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\) will save you a lot of time.

5. Graphing the Waves

If you take the height (Sine) or the horizontal distance (Cosine) from the Unit Circle and plot it as you go around, you get a beautiful wave pattern.

Key Terms for Graphs:

Amplitude: The height of the wave from the middle line to the peak. For \( y = \sin(x) \), the amplitude is 1.
Period: How long it takes for the wave to complete one full cycle and start repeating. For Sine and Cosine, the period is \(360^\circ\) (or \(2\pi\)).
Midline (Principal Axis): The horizontal line that runs through the exact middle of the wave.

Real-World Example: Think of a Ferris Wheel. Your height above the ground as the wheel turns is a Sine Graph. You start at the middle, go to the top, back to the middle, to the bottom, and back to the middle again.

6. Common Mistakes to Avoid

Even the best mathematicians make these slips! Keep an eye out for:

1. Calculator Mode: This is the biggest trap! Always check if your calculator is in DEG (Degrees) or RAD (Radians) mode before starting a problem.
2. Negative Signs: Remember the CAST diagram. If your angle is in the second quadrant (\(90^\circ\) to \(180^\circ\)), Cosine must be negative.
3. Confusing Sine and Cosine: Just remember: Cosine is the Cross (horizontal/x) and Sine is the Sky (vertical/y).

Summary: The Big Picture

Trigonometry in Year 5 is all about the relationship between angles and coordinates. By using the Unit Circle, we can find the Sine and Cosine of any angle, no matter how large. Whether you are using radians or degrees, the circle stays the same. Keep practicing the exact values and the CAST diagram, and you will master this chapter in no time! You've got this!

Welcome to the World of Vectors!

Welcome! Today, we are going to explore Vectors. If you have ever followed a map or played a video game where a character moves around the screen, you have already seen vectors in action. In this chapter, we will learn that a vector is more than just a number—it is a set of instructions that tells us how far to go and in which direction.

Don't worry if this seems a bit "out there" at first. By the end of these notes, you will see that vectors are just a handy way to organize information about movement and position.

1. What is a Vector? (The "What" and "Where")

In math, we usually deal with Scalars. A scalar is just a size (magnitude). For example, "5 kilograms" or "20 degrees Celsius" are scalars. They don't have a direction.

A Vector, however, has two parts: Magnitude (how big it is) and Direction (where it is going).

Analogy: If I tell you "Run 100 meters," that is a scalar. You might run in circles! But if I tell you "Run 100 meters East," that is a vector because I gave you a distance and a direction.

Key Terms to Remember:

Magnitude: The length or size of the vector.
Direction: The pathway along which the vector points.

Did you know? In the movie Despicable Me, the villain "Vector" explains his name by saying he has "both direction and magnitude!" He was actually being mathematically accurate.

2. How We Write Vectors (Notation)

There are a few ways to write a vector so we don't confuse it with a regular number:

1. Bold letters: Like v or a.
2. Underlined letters: Since we can't "write" in bold with a pen, we usually write \( \underline{v} \).
3. With an arrow: From point A to point B, written as \( \vec{AB} \).

The Column Vector

This is the most common way you will see vectors in Year 5. It looks like a tall bracket with two numbers inside:

\( \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} \)

The top number (x) tells you how many steps to move horizontally (right is positive, left is negative).
The bottom number (y) tells you how many steps to move vertically (up is positive, down is negative).

Example: \( \begin{pmatrix} 3 \\ -2 \end{pmatrix} \) means "Move 3 units right and 2 units down."

3. Adding and Subtracting Vectors

Adding vectors is actually very simple! You just add the top numbers together and the bottom numbers together. It's like adding two sets of instructions to get one final destination.

Adding Vectors

If \( \mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \), then:
\( \mathbf{a} + \mathbf{b} = \begin{pmatrix} 2+4 \\ 3+1 \end{pmatrix} = \begin{pmatrix} 6 \\ 4 \end{pmatrix} \)

Subtracting Vectors

Just like adding, but you subtract the components:
\( \mathbf{a} - \mathbf{b} = \begin{pmatrix} 2-4 \\ 3-1 \end{pmatrix} = \begin{pmatrix} -2 \\ 2 \end{pmatrix} \)

Quick Review: Think of vector addition as a "journey." If you follow the first set of instructions, and then follow the second set immediately after, where do you end up? That's your resultant vector.

4. Scalar Multiplication

This sounds fancy, but it just means "making the vector longer or shorter." When we multiply a vector by a regular number (a scalar), we multiply both the top and bottom numbers.

Example: If \( \mathbf{v} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \), then \( 2\mathbf{v} = \begin{pmatrix} 2 \times 3 \\ 2 \times -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix} \).

Multiplying by 2 makes the vector twice as long but keeps it pointing in the same direction.

Important Tip: If you multiply by a negative number, the vector flips and points in the exact opposite direction!

5. Magnitude (How long is the vector?)

Sometimes we need to know the exact length of the arrow. To find the magnitude of vector \( \mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix} \), we use a formula that looks exactly like Pythagoras' Theorem.

The symbol for magnitude is two vertical lines: \( |\mathbf{v}| \).
The formula is: \( |\mathbf{v}| = \sqrt{x^2 + y^2} \)

Step-by-Step Example:

Find the magnitude of \( \mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \):
1. Square the x value: \( 3^2 = 9 \)
2. Square the y value: \( 4^2 = 16 \)
3. Add them together: \( 9 + 16 = 25 \)
4. Take the square root: \( \sqrt{25} = 5 \)
The magnitude (length) is 5 units.

6. Common Mistakes to Avoid

- Mixing up x and y: Always remember—the top is "left/right" and the bottom is "up/down." Memory Aid: "X is a cross (across), Y is to the sky."
- Forgetting signs: If a vector says \( -3 \) on top, you must move left. If it says \( -3 \) on bottom, you must move down.
- Adding diagonally: Never add the top number to the bottom number. They must stay in their own "lanes."

7. Summary Checklist

Key Takeaway: Vectors are arrows that represent movement.
Key Takeaway: Column vectors use \( \begin{pmatrix} x \\ y \end{pmatrix} \) to show horizontal and vertical change.
Key Takeaway: To add vectors, add the "top" and "bottom" separately.
Key Takeaway: Use Pythagoras (\( \sqrt{x^2 + y^2} \)) to find the length (magnitude) of a vector.

Great job! You've just covered the essentials of vectors. Keep practicing drawing them on grid paper, and they will become second nature in no time!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a radian?

A radian is the angle created at the center of a circle when the length of the arc along the circumference is equal to the radius of the circle.

How do you convert an angle from degrees to radians?

To convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \).

How do you convert an angle from radians to degrees?

To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).

What are the key characteristics of the Unit Circle?

The Unit Circle is a circle with a radius of exactly 1 unit, centered at the origin \((0, 0)\) of a coordinate plane.

On the Unit Circle, which trigonometric functions represent the \(x\) and \(y\) coordinates?

The \(x\)-coordinate represents \(\cos(\theta)\) and the \(y\)-coordinate represents \(\sin(\theta)\).

How is the tangent function defined on the Unit Circle?

Tangent is defined as the ratio of the y-coordinate to the x-coordinate, or \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).

What is the fundamental Pythagorean identity in trigonometry?

The identity is \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

What does the CAST diagram represent?

The CAST diagram shows which trigonometric functions are positive in each of the four quadrants of the coordinate plane.

What are the exact values of Sine and Cosine for \(30^\circ\), \(45^\circ\), and \(60^\circ\)?

The exact values include \(\sin(30^\circ) = \frac{1}{2}\), \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), and \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\).

What is the amplitude and period of a basic Sine or Cosine wave?

For \(y = \sin(x)\) and \(y = \cos(x)\), the amplitude is 1 and the period is \(360^\circ\) (or \(2\pi\) radians).

What is the midline or principal axis in a trigonometric graph?

The midline is the horizontal line that runs exactly through the vertical center of the wave, halfway between the maximum and minimum values.

Why must you check your calculator mode before solving trigonometry problems?

Calculators calculate trigonometric values differently based on whether they are set to Degrees (DEG) or Radians (RAD).

What is a scalar in mathematics?

A scalar is a quantity that has only magnitude (size) and no direction.

What defines a vector?

A vector is a quantity that has both magnitude (size) and direction.

What is the magnitude of a vector?

Magnitude refers to the length or size of the vector.

What does the direction of a vector represent?

The direction represents the pathway or orientation along which the vector points.

What are the common ways to write a vector in notation?

Vectors are written using bold letters (v), underlined letters (\( \underline{v} \)), or an arrow over two points (\( \vec{AB} \)).

In a column vector, what does the top number represent?

The top number represents the horizontal change (left or right).

In a column vector, what does the bottom number represent?

The bottom number represents the vertical change (up or down).

How do you add two column vectors?

Add the top numbers together and the bottom numbers together separately.

How do you subtract one column vector from another?

Subtract the corresponding horizontal and vertical components separately.

What is a resultant vector?

A resultant vector is the single vector that represents the combined effect of two or more vectors.

What is scalar multiplication in the context of vectors?

Scalar multiplication is multiplying both the horizontal and vertical components of a vector by a single number.

What happens if you multiply a vector by a negative scalar?

The magnitude is scaled, and the direction of the vector is reversed (flipped).

What formula is used to calculate the magnitude of a vector \( \begin{pmatrix} x \\ y \end{pmatrix} \)?

The magnitude is calculated using \( |\mathbf{v}| = \sqrt{x^2 + y^2} \).

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