Exploring Trigonometric Functions with the Unit Circle

Understanding trigonometric functions using the unit circle is an important idea in Year 12 Mathematics.

When you learn how these functions come from the unit circle, it helps you see how sines, cosines, and tangents are linked together. This can also make solving problems easier.

What is the Unit Circle?

The unit circle is a special circle.

It has a radius of 1, and it sits right at the center of a coordinate plane.

The equation for the unit circle looks like this:

$x^2 + y^2 = 1$

Any point on the unit circle can be described using an angle called $\theta$. This angle is measured from the positive x-axis.

For a point $(x, y)$ on the circle, we can use the angle $\theta$ to figure out the coordinates like this:

$x = \cos(\theta)$ $y = \sin(\theta)$

What are Trigonometric Functions?

From the unit circle, we learn three main trigonometric functions:

1. Sine Function ($\sin(\theta)$): This tells us the y-coordinate of the point at angle $\theta$. It shows how far up or down the point is from the x-axis.

2. Cosine Function ($\cos(\theta)$): This tells us the x-coordinate of the point at the same angle. It shows how far left or right the point is from the y-axis.

3. Tangent Function ($\tan(\theta)$): This function comes from sine and cosine, and we can find it like this:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Tangent can be thought of as the slope of a line that forms with the x-axis. It shows the relationship between the opposite side and adjacent side in a right triangle.

How Trigonometric Ratios Change

As we spin a point around the unit circle, the values of sine, cosine, and tangent change in a repeating pattern.

Here’s how it works:

• Quadrantal Angles: At angles of $0^\circ$, $90^\circ$, $180^\circ$, and $270^\circ$, the sine and cosine reach their highest and lowest values. For example:
  • At $0^\circ$: $(x,y) = (1,0)$, so $\sin(0) = 0$ and $\cos(0) = 1$.
  • At $90^\circ$: $(x,y) = (0,1)$, so $\sin(90) = 1$ and $\cos(90) = 0$.
• In the First Quadrant: All the trigonometric ratios are positive. As we move from $0^\circ$ to $90^\circ$, $\sin(\theta)$ goes up while $\cos(\theta)$ goes down.

The Cycle of Trigonometric Functions

It’s easy to see that the sine and cosine functions repeat their values after one full turn (which is 360° or $2\pi$ radians):

• Periodicity: The sine and cosine functions cycle every $360^\circ$ or $2\pi$ radians:

$\sin(\theta + 360^\circ) = \sin(\theta)$ $\cos(\theta + 360^\circ) = \cos(\theta)$

This cycle goes on forever and helps us understand their use in different areas, like waves and vibrations.

Conclusion

To sum it up, the unit circle is a key part of understanding trigonometric functions.

By looking at the coordinates of points on the circle, students can better understand sine, cosine, and tangent.

Knowing how these ratios relate to each other makes tough math problems easier as students move forward in their studies.

Whether using graphs or seeing how it applies in real life, the unit circle shows us how beautiful and useful trigonometry can be in math.