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. The unit circle is a circle with a radius of one unit, centered at the origin $(0,0)$. 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, $x = \cos \theta$ represents the cosine function, which starts from the point $(0, 1)$. The cosine function has a maximum value of 1 at $\theta = 0^\circ, 360^\circ$ and a minimum value of -1 at $\theta = 180^\circ$, with a range of $-1 \leq \cos \theta \leq 1$.

The tangent function, represented by $\tan \theta$, is not continuous and breaks at $\theta = 90^\circ$ and $\theta = 270^\circ$, where it is undefined. It has no maximum or minimum values, with a range of $-\infty < \tan \theta < \infty$.

Exam Tips

• Understand the Unit Circle: Familiarize yourself with the unit circle's properties and how it relates to sine, cosine, and tangent functions. This will help in visualizing and solving trigonometric problems.

• Memorize Key Angles: Remember the key angles where sine, cosine, and tangent have specific values, such as $0^\circ, 90^\circ, 180^\circ, 270^\circ,$ and $360^\circ$.

• Graph Interpretation: Practice sketching and interpreting the graphs of $y = \sin \theta$, $y = \cos \theta$, and $y = \tan \theta$ for $0^\circ \leq \theta \leq 360^\circ$.

• Use Trigonometric Identities: Be comfortable using trigonometric identities to simplify expressions and solve equations.

• Practice Problem Solving: Work through example problems and quizzes to reinforce your understanding and application of the unit circle concepts.