Examples of cosecant, secant, cotangent functions

📚 Quick Study Guide

• 📐 Cosecant (csc θ): The reciprocal of sine. $csc \,\theta = \frac{1}{sin \,\theta} = \frac{hypotenuse}{opposite}$
• 📏 Secant (sec θ): The reciprocal of cosine. $sec \,\theta = \frac{1}{cos \,\theta} = \frac{hypotenuse}{adjacent}$
• ✏️ Cotangent (cot θ): The reciprocal of tangent. $cot \,\theta = \frac{1}{tan \,\theta} = \frac{adjacent}{opposite} = \frac{cos \,\theta}{sin \,\theta}$
• 💡 Key Identity: Remember that these functions are related to sine, cosine, and tangent, making them easy to derive if you know the primary trigonometric functions.
• 📈 Unit Circle: Visualizing these functions on the unit circle can help understand their behavior across different angles.

Practice Quiz

1. What is the value of $csc \,\theta$ if $sin \,\theta = \frac{1}{2}$?
   1. $\frac{1}{2}$
   2. $2$
   3. $0$
   4. $1$
2. If $cos \,\theta = \frac{\sqrt{3}}{2}$, what is the value of $sec \,\theta$?
   1. $\frac{2}{\sqrt{3}}$
   2. $\frac{\sqrt{3}}{2}$
   3. $2$
   4. $\frac{1}{2}$
3. Find $cot \,\theta$ when $tan \,\theta = 1$.
   1. $0$
   2. $1$
   3. $-1$
   4. $\infty$
4. What is the relationship between $csc \,\theta$ and $sin \,\theta$?
   1. $csc \,\theta = sin \,\theta$
   2. $csc \,\theta = \frac{1}{sin \,\theta}$
   3. $csc \,\theta = -sin \,\theta$
   4. $csc \,\theta = sin^2 \,\theta$
5. Determine $sec \,\theta$ if $cos \,\theta = 0.5$.
   1. $0.5$
   2. $1$
   3. $2$
   4. $0$
6. What is the value of $cot \,\theta$ if $tan \,\theta$ is undefined?
   1. $0$
   2. $1$
   3. Undefined
   4. $-1$
7. Given a right triangle, if the adjacent side is 4 and the opposite side is 3, what is the value of $cot \,\theta$?
   1. $\frac{3}{4}$
   2. $\frac{4}{3}$
   3. $\frac{5}{3}$
   4. $\frac{5}{4}$