Unit Circle

Summary and Exam Tips for Unit Circle

The Unit Circle is a subtopic of Geometry and Trigonometry, which falls under the subject Mathematics in the IB MYP curriculum. A unit circle is defined as a circle with a radius of one unit, centered at the origin of a coordinate plane. It is fundamental in understanding trigonometric functions. For any point $(x, y)$ on the unit circle, $y = \sin \theta$ represents the sine function. The sine function forms a wave starting from the origin, with $\sin \theta = 0$ at $\theta = 0^\circ, 180^\circ, 360^\circ$. Its maximum value is 1 at $\theta = 90^\circ$ and minimum value is -1 at $\theta = 270^\circ$, giving a range of $-1 \leq \sin \theta \leq 1$.

Similarly, the cosine function is represented by $x = \cos \theta$, starting from the point $(0, 1)$. The maximum value of $\cos \theta$ is 1 at $\theta = 0^\circ, 360^\circ$, and the minimum is -1 at $\theta = 180^\circ$, with a range of $-1 \leq \cos \theta \leq 1$.

The tangent function is unique as it is not continuous, breaking at $\theta = 90^\circ$ and $\theta = 270^\circ$ where it is undefined. It has no maximum or minimum values, with a range of $-\infty < \tan \theta < \infty$. Understanding these properties is crucial for graphing and solving trigonometric equations.

Exam Tips

• Understand the Unit Circle: Familiarize yourself with the unit circle's layout, including key angles and their corresponding sine, cosine, and tangent values.
• Memorize Key Values: Remember the maximum and minimum values of sine and cosine functions, and where tangent is undefined.
• Graph Interpretation: Practice sketching and interpreting graphs of $y = \sin \theta$, $y = \cos \theta$, and $y = \tan \theta$ over $0^\circ \leq \theta \leq 360^\circ$.
• Use Trigonometric Identities: Leverage identities like $\sin^2 \theta + \cos^2 \theta = 1$ to solve problems efficiently.
• Practice Problems: Solve various problems involving angles, radians, and trigonometric functions to build confidence and speed.