Quiz: Calculate all six trig values given a point (x,y).

📚 Quick Study Guide

• 📐 Definitions: Given a point $(x, y)$ on the terminal side of an angle in standard position, and letting $r = \sqrt{x^2 + y^2}$ be the distance from the origin to the point $(x, y)$, we define the trigonometric functions as follows:
  • sin($\theta$) = $\frac{y}{r}$
  • cos($\theta$) = $\frac{x}{r}$
  • tan($\theta$) = $\frac{y}{x}$ (where $x \neq 0$)
  • csc($\theta$) = $\frac{r}{y}$ (where $y \neq 0$)
  • sec($\theta$) = $\frac{r}{x}$ (where $x \neq 0$)
  • cot($\theta$) = $\frac{x}{y}$ (where $y \neq 0$)
• ➕ Finding r: Remember to calculate $r$ using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$. $r$ is always non-negative.
• 🧭 Quadrant Awareness: The signs of $x$ and $y$ determine the quadrant in which the angle lies, which affects the signs of the trigonometric functions.
• ✏️ Reciprocal Identities: Cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively. This can simplify calculations.

🧪 Practice Quiz

1. Given the point (3, 4), what is the value of sin($\theta$)?
   1. $\frac{3}{5}$
   2. $\frac{4}{5}$
   3. $\frac{3}{4}$
   4. $\frac{5}{4}$
2. Given the point (-5, 12), what is the value of cos($\theta$)?
   1. $\frac{12}{13}$
   2. $\frac{-5}{13}$
   3. $\frac{-5}{12}$
   4. $\frac{13}{-5}$
3. Given the point (8, -6), what is the value of tan($\theta$)?
   1. $\frac{4}{3}$
   2. $\frac{-3}{4}$
   3. $\frac{3}{4}$
   4. $\frac{-4}{3}$
4. Given the point (-1, -1), what is the value of csc($\theta$)?
   1. $\sqrt{2}$
   2. $-\sqrt{2}$
   3. $1$
   4. $-1$
5. Given the point (0, 5), what is the value of sec($\theta$)?
   1. $0$
   2. $1$
   3. Undefined
   4. $\frac{1}{5}$
6. Given the point (-2, 0), what is the value of cot($\theta$)?
   1. $0$
   2. Undefined
   3. $1$
   4. $-1$
7. Given the point ($\sqrt{3}$, 1), what is the value of sin($\theta$)?
   1. $\frac{1}{2}$
   2. $\frac{\sqrt{3}}{2}$
   3. $\frac{1}{\sqrt{3}}$
   4. $2$