Chapter 6 Test Review: Trigonometry and Its Applications

Hello Math Students! This review is designed to help you ace your upcoming Chapter 6 test. Let's dive into the key concepts and practice problems to boost your confidence!

Key Concepts

  • Trigonometric Functions: Understanding sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent) is crucial. Remember SOH CAH TOA!
  • Unit Circle: Familiarize yourself with the unit circle and the values of trigonometric functions at key angles (e.g., 0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$).
  • Graphing Trigonometric Functions: Be able to graph sine, cosine, and tangent functions, and identify their amplitude, period, phase shift, and vertical shift.
  • Inverse Trigonometric Functions: Understand the concept of inverse trigonometric functions and how to evaluate them.
  • Applications of Trigonometry: Solve real-world problems involving right triangles, angles of elevation/depression, and simple harmonic motion.

Practice Problems and Examples

1. Area of a Sector

The area of a sector of a circle can be found using the formula: $A = \frac{1}{2}r^2\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.

Example: A windshield wiper clears a sector with a radius of 81 cm and an angle of 130°. To find the area cleared, we first convert 130° to radians: $130 \times \frac{\pi}{180} = \frac{13\pi}{18}$. Then, $A = \frac{1}{2}(81)^2(\frac{13\pi}{18}) \approx 7443.2 \text{ cm}^2$. Similarly, if the radius is 32 cm, $A = \frac{1}{2}(32)^2(\frac{13\pi}{18}) \approx 1161.7 \text{ cm}^2$.

2. Evaluating Trigonometric Functions

Given $\theta = 135°$, let's find the exact values of some trigonometric expressions:

  • $\cos(-\theta) = \cos(-135°) = -\frac{\sqrt{2}}{2}$
  • $\cos^2(\theta) = (\cos(135°))^2 = (-\frac{\sqrt{2}}{2})^2 = \frac{1}{2}$
  • $\cos(2\theta) = \cos(270°) = 0$

3. Finding Trigonometric Values Beyond 360 Degrees

To find $\tan(960°)$, subtract multiples of 360° until you get an angle between 0° and 360°: $960° - 360° - 360° = 240°$. Then, $\tan(240°) = \sqrt{3}$. Similarly, $\csc(-\frac{5\pi}{4}) = \sqrt{2}$.

4. Analyzing Trigonometric Functions

Consider the function $y = 4\sin(\pi x + \frac{\pi}{4}) + 1$. We can rewrite this as $y = 4\sin(\pi(x + \frac{1}{4})) + 1$. The amplitude is 4, the period is $\frac{2\pi}{\pi} = 2$, and the phase shift is $-\frac{1}{4}$.

5. Modeling with Trigonometric Functions

Suppose the population of an animal species is modeled by $p(t) = 3050 + 1250\cos(1.5t)$. The maximum population is $3050 + 1250 = 4300$, and the minimum population is $3050 - 1250 = 1800$. The period of the function is $\frac{2\pi}{1.5} \approx 4.19$ years. The number of cycles per year is approximately $\frac{1}{4.19} \approx 0.24$.

Good luck with your test! Remember to review your notes, practice problems, and stay confident. You got this!