Chapter 7 & 8 Test Review

Welcome to the review for your upcoming Chapter 7 and 8 test! This guide will help you refresh your knowledge and build confidence. Remember to practice and don't hesitate to ask questions!

Law of Sines and Cosines

These are crucial tools for solving triangles that aren't necessarily right triangles. Let's break them down:

  • Law of Sines: This law states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant. Mathematically, it's expressed as: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
  • Law of Cosines: This law relates the lengths of the sides of a triangle to the cosine of one of its angles. It's particularly useful when you know two sides and the included angle (SAS) or all three sides (SSS). The formulas are:
    • $a^2 = b^2 + c^2 - 2bc \cos A$
    • $b^2 = a^2 + c^2 - 2ac \cos B$
    • $c^2 = a^2 + b^2 - 2ab \cos C$

Example: Consider a triangle ABC where $c = 67$, $a = 28$, and $C = 105^\circ$. We can use the Law of Sines to find angle A: $$\frac{\sin A}{a} = \frac{\sin C}{c} \implies \sin A = \frac{a \sin C}{c} = \frac{28 \sin 105^\circ}{67}$$ Then, $A = \sin^{-1}(\frac{28 \sin 105^\circ}{67}) \approx 23.8^\circ$. We can find angle B using $180^\circ - 105^\circ - 23.8^\circ = 51.2^\circ$. Use the law of sines again to find side b.

Another Example: A rover on Mars travels 22 km south and then adjusts its course 51° westward for 55 km. We can find the distance and direction back to the starting point using the Law of Cosines and Sines.

Area of a Triangle

You can calculate the area of a triangle using the following formulas:

  • If you know the base ($b$) and height ($h$): $$A = \frac{1}{2}bh$$
  • Using trigonometry: $$\text{Area} = \frac{1}{2}ab \sin(C)$$, where $a$ and $b$ are two sides and $C$ is the included angle.

Example: If $b = 9$ and $h = 4 \sin(35)$, then the area is $A = \frac{1}{2}(9)(4 \sin(35)) \approx 10.3 \text{ ft}^2$

Trigonometric Equations and Identities

Mastering trigonometric equations and identities is vital. Here are some key concepts:

  • Identities: These are equations that are true for all values of the variable. Examples include:
    • $\sin^2(x) + \cos^2(x) = 1$
    • $1 + \cot^2(x) = \csc^2(x)$
    • $1 + \tan^2(x) = \sec^2(x)$
  • Solving Trigonometric Equations: Use algebraic techniques and trigonometric identities to isolate the trigonometric function. Remember to consider all possible solutions within the given interval (e.g., $[0, 2\pi)$).

Example: Solve $\cot(x) = 0$ in the interval $[0, 2\pi)$. The solutions are $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$.

Sum and Difference Formulas

These formulas allow you to find trigonometric functions of sums or differences of angles.

  • $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
  • $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
  • $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

Product-to-Sum Formulas

These formulas rewrite products of trigonometric functions as sums or differences.

  • $\cos(\theta)\cos(\phi) = \frac{[\cos(\theta - \phi) + \cos(\theta + \phi)]}{2}$

Example: Rewrite $\cos(5d)\cos(3d)$ as a sum: $$\frac{\cos(2d) + \cos(8d)}{2}$$

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