Welcome to Chapter 7, Lesson 1!

In this lesson, we'll be exploring a variety of trigonometric concepts. Get ready to expand your knowledge and skills in trigonometry!

Important Resources

  • Team Meeting Link: Join our virtual meetings for live discussions and Q&A sessions.
  • Trig Cheat Sheet: A handy reference guide for trigonometric identities and formulas (see attached PDF).
  • Topic List: A structured overview of the topics covered in this chapter.

Key Concepts and Skills

Here's a breakdown of what we'll be covering:

1. Inverse Trigonometric Values

  • Using a calculator to approximate inverse trigonometric values (e.g., $\sin^{-1}(x)$, $\cos^{-1}(x)$, $\tan^{-1}(x)$). Remember that the domain and range of inverse trig functions are important. For example, $\sin^{-1}(x)$ has a domain of $[-1,1]$ and a range of $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

2. Finding Trigonometric Function Values

  • Finding values of trigonometric functions given information about an angle (Problem type 2 & 3). This often involves using the definitions of sine, cosine, tangent, cosecant, secant, and cotangent in terms of the sides of a right triangle, or the coordinates of a point on the unit circle.

3. Simplifying Trigonometric Expressions

  • Simplifying trigonometric expressions using fundamental identities. Key identities include: $\sin^2(\theta) + \cos^2(\theta) = 1$, $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, and $\sec(\theta) = \frac{1}{\cos(\theta)}$.

4. Verifying Trigonometric Identities

  • Verifying trigonometric identities by manipulating one side of the equation to match the other.

5. Proving Trigonometric Identities

  • Proving trigonometric identities using various techniques (Problem types 1, 2, & 3). This often requires creative use of algebraic manipulation and trigonometric identities.

6. Sum and Difference Identities

  • Applying sum and difference identities to simplify expressions and solve equations (Problem types 1 & 2). Examples include: $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ and $\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$.

7. Proving Identities with Sum and Difference Properties

  • Proving trigonometric identities using sum and difference properties (Problem types 1 & 2).

8. Double and Half-Angle Identities

  • Using double-angle identities (Problem type 1): $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$.
  • Using half-angle identities (Problem type 1): $\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}}$, $\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$. Remember to consider the quadrant of $\frac{\theta}{2}$ to determine the correct sign.

Remember to practice consistently and refer to the Trig Cheat Sheet for quick access to formulas and identities. Good luck!