Hi Everyone!

On this page you will find some material about Lesson 32. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

**Lesson 32: Law of Sines **

Table of Contents

### Resources

*In this section you will find some important information about the specific resources related to this lesson: *

*the learning outcomes,**the section in the textbook,**the WeBWorK homework sets,**a link to the pdf of the lesson notes,**a link to a video lesson.*

**Learning Outcomes** (from Coburn and Herdlick’s Trigonometry book)

- Develop the law of sines and use it to solve ASA and AAS triangles.
- Solve SSA triangles (the ambiguous case) using the law of sines.
- Use the law of sines to solve applications.

**Topic**. This lesson covers Section 7.1: Oblique Triangles and the Law of Sines.

**WeBWorK**. There is one WeBWorK assignment on today’s material:

LawOfSines

**Lesson Notes.**

These notes are used in Lessons 32 and 33. Today’s lesson is on pages 1-3.

**Video Lesson.**

Video Lesson 32 (based on Lesson 32 Notes)

This video is used in Lessons 32 and 33. For today’s lesson, watch from [0:00] to [8:01].

### Warmup Questions

*These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.*

#### Warmup Question 1

If $\cos \theta = a$, what is $\arcsin a$?

#### Show Answer 1

$\arcsin a = \theta$

#### Warmup Question 2

Find $\arcsin 0.766$ in degrees.

#### Show Answer 2

$\arcsin 0.766=50^{\circ}$

### Review

*If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.*

Need a review? Check

### Quick Intro

*This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.*

**A Quick Intro to Law of Sines **

**Key Words.** SSS, SAS, AAS, ASA, Oblique triangle, solving a triangle, law of sines

$\bigstar$ We denote the sides of a tringle $\Delta ABC$ by $a$, $b$ and $c$ as the sides opposite to the angles $A$, $B$ and $C$, respectively.

$\bullet$ SSS means that the three sides are known.

$\bullet$ SAS means that two sides and the adjacent angle are known.

$\bullet$ AAS means that two angles and one side (not between the two angles) are known.

$\bullet$ ASA means that two angles and the adjacent side are known.

$\bigstar$ **Solving a triangle** means to find the unknown sides and angles.

$\bigstar$ A triangle that does not have a right angle is called **oblique**.

$\bigstar$ In a right triangle, you use the trig ratios to solve it. Otherwise, the triangle is oblique in which case consider:

$\bigstar$ **Law of sines** (for ASA/AAS triangles)

\[\dfrac{a}{\sin A} = \dfrac{b}{\sin B}=\dfrac{c}{\sin C}.\]

### Video Lesson

*Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!*

**Video Lesson**

This video is used in Lessons 32 and 33. For today’s lesson, watch from [0:00] to {8:01].

**A description of the video**

In the video you will see the following problems.

- Given a triangle whose angles are $40^{\circ}$, $60^{\circ}$ and $80^{\circ}$ where the sides opposite to $40^{\circ}$, $60^{\circ}$ and $80^{\circ}$ measure 5, $x$ and $y$, respectively, find $x$ and $y$.
- Given a triangle whose angles are $\theta$ and $95^{\circ}$, and the sides opposite to $\theta$ and $95^{\circ}$ measure 5 and $10$, respectively, find $\theta$.

### Try Questions

*Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.*

#### Try Question 1

Solve the triangle having the following properties: side $b$ = 385 m, $\angle C = 67^{\circ}$, and side $a$ =490 m.

#### Show Answer 1

By the law of cosines,

$$c^2=a^2+b^2-2ab\cos C$$

$$c^2 = (490)^2+(385)^2-2\cdot 490\cdot 385\cdot\cos 67^{\circ}$$

$$c^2 \approx 240902.1452$$

$$c\approx 490.82$$

By the law of sines,

$$\dfrac{\sin A}{a}=\dfrac{\sin C}{c}$$

$$\dfrac{\sin A}{490} = \dfrac{\sin 67^{\circ}}{490.82}$$

$$\sin A =\dfrac{490\cdot \sin 67^{\circ}}{490.82} $$

$$\sin A\approx 0.91896699 $$

$$A\approx \arcsin (0.91896699)\approx 66.77 $$

Hence $B= 180-A-C \approx 180 – 66.77 – 67 =46.23$.

$\bullet$ **sides:**

$a=490$

$b=385$

$c\approx 490.82$

$\bullet$ **angles:**

$A\approx66.77^{\circ}$

$B\approx 46.23^{\circ}$

$C=67^{\circ}$

### WeBWorK

*You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.*

**WeBWork**

It is time to do the homework on WeBWork:

LawOfSines

When you are done, come back to this page for the Exit Questions.

### Exit Questions

*After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!*

#### Exit Questions

- Why is there not an SSA theorem?
- When can you use the law of sines?
- Why is it better to use more accuracy or exact answers in early parts of solving triangles?

$\bigstar$ Solve the triangle $\triangle ABC$ for which $\angle A=31^{\circ}$, $c=207\;m$, and $b=250\;m$.

#### Show Answer

### Need more help?

*Don’t wait too long to do the following.*

- Watch the additional video resources.

- Talk to your instructor.
- Form a study group.
- Visit a tutor. For more information, check the tutoring page.