Introduction to the Unit Circle and Trigonometric Identities
Welcome to one of the most visual and useful parts of the Higher Level (HL) Geometry and Trigonometry topic! In earlier years, you learned about SOH CAH TOA and how trigonometry works inside right-angled triangles. But what happens if the angle is larger than \(90^\circ\)? What if the angle is negative?
In this chapter, we move beyond the triangle and into the Unit Circle. This allows us to define trigonometry for any angle imaginable. We will also look at how to use "identities" (mathematical rules that are always true) to simplify expressions and solve equations using your Graphic Display Calculator (GDC). Don't worry if it feels abstract at first—once you see the symmetry, it all clicks together!
1. The Unit Circle Definitions
The Unit Circle is simply a circle with a radius of \(1\) unit, centered at the origin \((0,0)\) on a coordinate plane.
Imagine a point \(P\) moving around this circle. The angle \(\theta\) (theta) is measured from the positive \(x\)-axis, turning counter-clockwise. Because the radius is \(1\), the coordinates of point \(P(x, y)\) are defined by the trigonometric functions:
- The x-coordinate is \(\cos \theta\)
- The y-coordinate is \(\sin \theta\)
- Therefore, any point on the circle is \(( \cos \theta, \sin \theta )\)
Understanding the Quadrants (CAST)
Depending on which "slice" of the circle the angle falls into, the values of sine and cosine can be positive or negative:
- Quadrant I (\(0^\circ\) to \(90^\circ\)): All values (\(\sin, \cos, \tan\)) are Positive.
- Quadrant II (\(90^\circ\) to \(180^\circ\)): Only Sine is positive.
- Quadrant III (\(180^\circ\) to \(270^\circ\)): Only Tangent is positive.
- Quadrant IV (\(270^\circ\) to \(360^\circ\)): Only Cosine is positive.
Memory Trick: Remember the acronym CAST (starting from Quadrant IV and going around) or All Stations To Central (starting from Quadrant I).
Key Takeaway
The unit circle expands trig beyond triangles. Just remember: Cos is x, Sin is y.
2. Fundamental Trigonometric Identities
An identity is an equation that is true for every possible value of \(\theta\). There are two specific identities you must know for the AI HL course:
A. The Tangent Identity
\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
This comes from the fact that in the unit circle, \(\tan \theta\) represents the gradient (slope) of the line from the origin to point \(P\). Since \(gradient = \frac{rise}{run} = \frac{y}{x}\), and we know \(y = \sin \theta\) and \(x = \cos \theta\), the identity follows naturally.
B. The Pythagorean Identity
\(\cos^2 \theta + \sin^2 \theta = 1\)
This is just the Pythagorean theorem (\(a^2 + b^2 = c^2\)) applied to the unit circle! Since the horizontal side of the triangle is \(\cos \theta\), the vertical side is \(\sin \theta\), and the hypotenuse (radius) is \(1\), we get \(( \cos \theta )^2 + ( \sin \theta )^2 = 1^2\).
Common Mistake: Be careful with notation. \(\cos^2 \theta\) means \((\cos \theta)^2\). It does not mean \(\cos(\theta^2)\).
Quick Example
If you are told that \(\sin \theta = 0.6\) and the angle is in the second quadrant, you can find \(\cos \theta\) using the identity:
\(\cos^2 \theta + (0.6)^2 = 1\)
\(\cos^2 \theta + 0.36 = 1\)
\(\cos^2 \theta = 0.64\)
\(\cos \theta = \pm 0.8\)
Since it is in the second quadrant (where cosine is negative), \(\cos \theta = -0.8\).
3. The Ambiguous Case of the Sine Rule
In your previous study of the Sine Rule (\(\frac{a}{\sin A} = \frac{b}{\sin B}\)), you usually found one unique answer. However, there is a special situation called the Ambiguous Case (SSA - Side, Side, Angle).
This happens when you are given two sides and a non-included angle. Because \(\sin \theta = \sin(180^\circ - \theta)\), your calculator might give you an acute angle, but an obtuse angle could also be mathematically possible.
When do two triangles exist?
Imagine you have side \(a\), side \(b\), and angle \(A\). Two triangles are possible if:
- The side opposite the angle (\(a\)) is shorter than the adjacent side (\(b\)).
- The side opposite the angle (\(a\)) is longer than the vertical height of the triangle (\(b \sin A\)).
How to solve it:
Step 1: Use the Sine Rule to find the first possible angle: \(\theta_1\).
Step 2: Find the second possible angle: \(\theta_2 = 180^\circ - \theta_1\).
Step 3: Check if \(\theta_2\) is possible by adding it to your original given angle. If the sum is less than \(180^\circ\), you have two valid triangles!
Key Takeaway
If you see a "Side-Side-Angle" problem, always check if a second, obtuse triangle could exist using \(180^\circ - \theta\).
4. Solving Trigonometric Equations Graphically
In the Applications and Interpretation (AI) course, the focus is on using technology. You are rarely required to solve complex trig equations by hand. Instead, you will solve them in a finite interval (e.g., \(0 \le \theta \le 360^\circ\) or \(0 \le \theta \le 2\pi\)).
Step-by-Step GDC Method
To solve an equation like \(3 \sin(2x) = 1.5\) for \(0^\circ \le x \le 180^\circ\):
- Check your mode: Ensure your calculator is in Degrees or Radians as required by the question. (Note: Radians are covered in sub-topic 3.7).
- Enter the functions:
Let \(y_1 = 3 \sin(2x)\)
Let \(y_2 = 1.5\) - Set the Window: Manually set your \(x_{min}\) to \(0\) and \(x_{max}\) to \(180\). Use "Zoom Fit" or adjust \(y\) to see the waves clearly.
- Find Intersections: Use the Intersect command (usually 2nd > Calc > Intersect or G-Solv > Intse) to find every point where the graphs cross.
Why is this better than algebra?
Trig functions repeat forever. A calculator graph helps you visually see exactly how many solutions exist within your specific interval so you don't miss any!
Key Takeaway
Don't struggle with algebra if the question allows a graphical approach. Sketch the two sides of the equation and find where they meet.
Chapter Summary
- The Unit Circle: Radius is \(1\). \((x, y) = (\cos \theta, \sin \theta)\).
- CAST Diagram: Helps determine if \(\sin, \cos\), or \(\tan\) is positive in a specific quadrant.
- Pythagorean Identity: \(\cos^2 \theta + \sin^2 \theta = 1\). Very useful for switching between \(\sin\) and \(\cos\).
- Ambiguous Case: In SSA situations, there might be two possible triangles (one acute, one obtuse).
- GDC is King: For equations, graph both sides and find the intersections within the given domain.