## Law of Cosines

#### Aligned To Common Core Standard:

**High School Geometry** - HSG-SRT.D.10

What is the Law of Cosines?
When it comes to solving triangles, we have two different types of cases to deal with. The first involves right-angled triangles and the second case is when dealing with triangles other than right-angled triangles. To solve right-angled triangles, we use Pythagoras theorem and trigonometric functions including sin, cos, and tan. For solving isosceles, scalene, or equilateral triangles, we apply the Sine and Cosine rules.
For the sake of this topic, we will focus on only one of the rules. The cosine rule is given by; a^{2} = b^{2} + c^{2} - 2bc cosA
The law of cosine is used when two sides and the angle between them is provided and we have to find the third side. It is also used in a situation where all three sides are given, and the angle are unknown.
This series of worksheets and lessons helps students learn how to use the cosine rule which allows us to learn unknown values in triangles.

### Printable Worksheets And Lessons

- Find a Side Step-by-Step Lesson- Find the length of side A and round it. See if you can apply this skill quickly.
- Guided Lesson - I am a little obsessed with the rhombus in these. You will find missing measures of length and angles.
- Guided Lesson Explanation - I find these to be a little more difficult than I would expect for most students.
- Practice Worksheet - These might take students a bit longer to complete because they are packed with info.
- Matching Worksheet - Match the lengths of the sides to the problems that they are derived from.
- Cosine Worksheet Five Pack - Find that side measurement please. There are ten problems on each of the pages.
- Find the Cosine of One Point Worksheet Five Pack - We draw this back a bit from the calculator.

#### Homework Sheets

Find the length of sides and diagonals in these.

- Homework 1 - In triangle XYZ, side y = 8, side z = 4 and m
- Homework 2 - In a rhombus whose side measures 15 and the smaller angle is 130°; find the length of the larger diagonal, to the nearest tenth.
- Homework 3 - Find the missing side of the triangle. We would recommend that you use the technique we have explored here.

#### Practice Worksheets

Usually you will see these answers in decimals to the nearest tenths on assessments.

- Practice 1 - In a rhombus whose side measures 51 and the smaller angle is 160°; find the length of the larger diagonal, to the nearest tenth.
- Practice 2 - How long is that diagonal side? Guess how to set these problems up.
- Practice 3 - You can use this skill to learn a great deal about triangles.

#### Math Skill Quizzes

The tests really pack a great deal of information in there.

- Quiz 1 - Start with the equation: b
^{2}= a^{2}+ c^{2}- 2ac Cos B - Quiz 2 - If you know the measure of an angle, you can learn a lot about the end value.
- Quiz 3 - Find side 'p' to the nearest integer. It would work through all of the different areas.

### How Does This Skill Apply to The Real World?

When students first start with this skill, they think these types of exercise are just logic problems that have little or no relevance to real life. The truth is that you can use math like this to create model of many real world things and it can be used to make many calculations that can help you make many positive decisions in your everyday life. Think about it for a minute, anything that forms a triangle where you do not have the ability to measure this skill applies. One super relative application is to determine places that are closer in distance to you. For example, my friend Carlos works two jobs. He works out in the morning at the gym and needs to determine if he should go home or to friend's house to shower before work because the gym's water is out. He can model this situation with a triangle. Carlos would be one point of the triangle, his friend's house another, and his home the last point. Carlos can then apply this skill to figure out which of the two locations would be the most convenient for him.