Solved examples using basic trigonometric identities

📚 Quick Study Guide

• 📐 Basic Trigonometric Ratios:
  • Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
  • Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$
• 🔄 Reciprocal Identities:
  • Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$
• ➕ Pythagorean Identities:
  • $\sin^2(\theta) + \cos^2(\theta) = 1$
  • $1 + \tan^2(\theta) = \sec^2(\theta)$
  • $1 + \cot^2(\theta) = \csc^2(\theta)$
• ➗ Quotient Identities:
  • $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
  • $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$
• ➖ Even and Odd Identities:
  • $\sin(-\theta) = -\sin(\theta)$
  • $\cos(-\theta) = \cos(\theta)$
  • $\tan(-\theta) = -\tan(\theta)$

Practice Quiz

1. What is the value of $\sin^2(\theta) + \cos^2(\theta)$?
   1. 0
   2. 1
   3. 2
   4. -1
2. If $\sin(\theta) = \frac{3}{5}$, what is the value of $\csc(\theta)$?
   1. $\frac{4}{5}$
   2. $\frac{5}{3}$
   3. $\frac{4}{3}$
   4. $\frac{5}{4}$
3. Which of the following is equal to $\tan(\theta)$?
   1. $\frac{\cos(\theta)}{\sin(\theta)}$
   2. $\frac{1}{\cot(\theta)}$
   3. $\frac{1}{\sec(\theta)}$
   4. $\frac{\sin(\theta)}{\cos(\theta)}$
4. Simplify: $(1 + \tan^2(\theta)) \cdot \cos^2(\theta)$?
   1. $\sin^2(\theta)$
   2. $\cos^2(\theta)$
   3. 1
   4. $\tan^2(\theta)$
5. If $\cos(\theta) = \frac{1}{2}$, what is the value of $\sec(\theta)$?
   1. 2
   2. $\frac{1}{2}$
   3. $\sqrt{3}$
   4. $\frac{\sqrt{3}}{2}$
6. Which identity is equivalent to $\csc^2(\theta) - 1$?
   1. $\sin^2(\theta)$
   2. $\cos^2(\theta)$
   3. $\tan^2(\theta)$
   4. $\cot^2(\theta)$
7. Given $\tan(\theta) = 1$, find the value of $\cot(\theta)$.
   1. 0
   2. 1
   3. -1
   4. 2