Trigonometry - Unit Circle and Function Extensions

Unit Circle and Common Trigonometric Values Understanding the Unit Circle The unit circle is a circle with radius 1 centered at the origin $(0,0)$ in the coordinate plane. It serves as the fundamental tool for defining trigonometric functions and understanding their behavior. Why the Unit Circle Matters The unit circle connects angle measures to coordinates. When you place an angle $\theta$ in standard position (vertex at the origin, initial side along the positive $x$-axis, terminal side rotated counterclockwise), the point where the terminal side intersects the unit circle has coordinates: $$(x, y) = (\cos\theta, \sin\theta)$$ This is the key insight: the cosine of an angle gives the $x$-coordinate, and the sine of an angle gives the $y$-coordinate on the unit circle. This definition automatically tells us that for any angle: $\sin^2\theta + \cos^2\theta = 1$ (since points lie on a circle of radius 1) Both $\sin\theta$ and $\cos\theta$ are always between $-1$ and $1$ Common Angle Values You need to know the sine and cosine values for certain key angles. These appear constantly in trigonometry, and memorizing them (or understanding how to derive them) is essential. The Key Four Angles At $0°$: The terminal side points along the positive $x$-axis, so the point is $(1, 0)$. $$\sin 0° = 0, \quad \cos 0° = 1, \quad \tan 0° = 0$$ At $45°$: This angle creates an isosceles right triangle. By symmetry, the point is at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. $$\sin 45° = \frac{\sqrt{2}}{2}, \quad \cos 45° = \frac{\sqrt{2}}{2}, \quad \tan 45° = 1$$ At $60°$: Using properties of a 30-60-90 triangle, the point is $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. $$\sin 60° = \frac{\sqrt{3}}{2}, \quad \cos 60° = \frac{1}{2}, \quad \tan 60° = \sqrt{3}$$ At $30°$: This is the complement of $60°$. The point is $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$. $$\sin 30° = \frac{1}{2}, \quad \cos 30° = \frac{\sqrt{3}}{2}, \quad \tan 30° = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$ How to Remember These Values Rather than pure memorization, understand the patterns: At $45°$: sine and cosine are equal (both $\frac{\sqrt{2}}{2}$) At $30°$ and $60°$: notice that $\sin 30° = \cos 60°$ and $\sin 60° = \cos 30°$ (these angles are complementary) The $\sqrt{3}$ and $\sqrt{2}$: appear in triangles with special angle ratios A common memory aid is the "special triangles" approach: understand that 30-60-90 and 45-45-90 triangles have fixed side ratios, which directly give you these trig values. Extension to All Real Angles Once you understand the unit circle for common angles between $0°$ and $90°$, you can extend these definitions to any angle—positive, negative, or larger than $360°$. For angles in other quadrants, use the reference angle (the acute angle between the terminal side and the $x$-axis) along with the signs of $x$ and $y$ in that quadrant: Quadrant I: both sine and cosine are positive Quadrant II: sine is positive, cosine is negative Quadrant III: both are negative Quadrant IV: sine is negative, cosine is positive For example, $\sin 150° = \sin 30° = \frac{1}{2}$ (since the reference angle is $30°$ and sine is positive in Quadrant II), while $\cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}$ (cosine is negative in Quadrant II). <extrainfo> Complex Extension For advanced courses, trigonometric functions extend to complex arguments using Euler's formula: $$e^{iz} = \cos z + i\sin z$$ where $z$ is a complex number and $i$ is the imaginary unit. From this, sine and cosine can be expressed in terms of exponentials: $$\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \quad \cos z = \frac{e^{iz} + e^{-iz}}{2}$$ These formulas allow trigonometric functions to be evaluated at complex arguments, which is important in advanced mathematics, engineering, and physics. However, this topic is typically beyond introductory trigonometry and may not appear on your exam. </extrainfo>