# Online Trigonometry Tutoring: Laws of sines & cosines

Home | Online Math Tutoring | Online Trigonometry Tutoring | **What Is The Law Of Sines?**

**Relationship between the angles and the side lengths of triangle**

**Simplify trigonometry. **Our expert and **qualified trigonometry tutors** understand your problems and guide you towards better grades in trigonometry. Our aim is to help the student understand concepts in trigonometry and master the techniques of solving problems quickly with confidence.

## Learn the laws of sines and cosines from certified online trigonometry tutor

**✓** It states that “When we divide side a by the sine of angle A, it is equal to side b divided by the

sine of angle B, and also equal to side c divided by the sine of angle C”.

**✓** It can be used to –

Calculate the unknown sides (Triangulation)

**✓** Angles of a Triangle Law can be applied if

- SSA – two sides and angle not included between them are given
- ASA – two angles and side between them are given
- SAS – two angles and one side that is not included in the angles

#### What is the Law of Cosines

**Relationship between the side lengths and the angles of a triangle**

The Law of Cosines can be given as

Other two versions of Law of Cosines are:

*a*^{2} = *b*^{2} + *c*^{2} – 2 *bc* cos *A*

*b*^{2} = *a*^{2} + *c*^{2} – 2*ac* cos *B*

#### What are the Uses of Laws of Cosines?

The Laws of Cosines is used:

- To find the 3rd side of a triangle when we know the 2 sides and the angle between them (
**SAS**) - To find the angles of a triangle when we know all the 3 sides (
**SSS**) of triangle.

#### When should we use the Laws of Sines?

1.If you are given: Angle-Side-Angle (**ASA**) or Angle-Angle-Side (**AAS**)*. OR*

2.If you are given: Hypotenuse-Leg (**HL**), you have a *right triangle*. OR

3.If you are given: Side-Side-Angle (**SSA**– in that order!!), then you have the __ AMBIGUOUS CASE__ ( we’ll discuss this case after the examples)

**Example 1**

**In a triangle ***ABC***, ***a*** = 10, ***b*** = 5, and ∠***A ***= 45****°****. Find the value of ∠***B***.**

* a*/sin *A* = *b*/sin *B*

10/sin 45 = 5/sin *B*

sin B = 1/2√2 or √2/4 (because sin 45 = 1/√2)

sin *B* = 0.3535

* *B = sin^{-1}(0.3535) = 20.7

**Example 2**

**In triangle ***ABC***, side ***b*** = 5 cm, ***c*** = 10 cm, and the angle at ***A*** is 60°. Find side ***a***.**

According to law of cosines,* a*^{2} = *b*^{2} + *c*^{2} – 2*bc* cos 60

* * a^{2} = 5^{2} + 10^{2} – 2 x 5 x 10 cos 60

* a*^{2} = 125 – 2 x 5 x 10 x ½ (cos 60 = ½)

= 125 – 50

* a*^{2} = *75*

* a* = √75

**Ambiguous Case – Law of ****Sines**

There are 5 situations when we need to use the Ambigious Case of the Law of Sines:

**Case I:** Angle is acute.

Side ‘*a*’ may or may not be long enough

to reach side ‘*c*’. We calculate the height

of the altitude from angle C to side *c* to

compare it with side *a*.

**Using Case I: **First, use SOH-CAH-TOA to find *h*:

sinA =

h = bsin A

**Then, compare ‘***h***’ to sides ***a*** and ***b*** . . .**

**Case I: **If *a* < *h*, then NO triangle exists with these dimensions.

sinA =

h = bsin A

**Then, compare ‘***h***’ to sides ***a*** and ***b*** . .**

**Case II: **If *h* < *a* < *b*, then TWO triangles exist with these dimensions.

If we open side ‘*a*’ to the outside of *h*, If we open side ‘*a*’ to the inside of *h*, angle B is obtuse.

angle B is acute.

**Case III: **If *h* < *b* < *a*, then ONE triangle exists with these dimensions.

Since side *a* is greater than side *b*, side *a* cannot open to the inside of *h*, it can only open to the outside, so there is only 1 triangle possible!

**Case IV: **If *h* = *a*, then ONE triangle exists with these dimensions.

If *a* = *h*, then angle B must be a right angle and there is only one possible triangle with these dimensions**.**

Given a triangle with angle A = 30°, side *a = *14 cm and side *b *= 15 cm, find the other dimensions.

Using Law of Sines, =

= 10sinB = 14 sin30

sinB = = 0.7

B = (0.7) = 44.42 ≅ 44

Angles could be 30°, 44°, and 106°: sum 180°.

The angle from Quadrant II could create angles 30°, 14°, and 136°: sum 180°.

**Check Point**

**Problem 1.** *A* = 40; *B* = 20; *a* = 2. Find side *b*.

**Problem 2.** If *b* = 5, *c* = 2, *A *= 30, find *a*.

**Problem 3.** Given *ABC,* *a* = 8, *b* = 5,*c* = 7, find *C* using law of cosines.

##### Answer Key

*b*= 3.76 approx*a*= 3.42 approx*C*= 60 degrees

Play with triangles the eTutorWorld way! Make your trigonometry tutoring sessions more fun and interactive. You will have an experienced and certified tutor who will help you succeed. We aim to effectively develop and sustain high-quality out-of-school learning and deliver a curriculum that each student will enjoy. Be it **homework help** with trigonometry or practice worksheets for your math hour at home, eTutorWorld has it all!

It’s really magical yet triangular. Find out for yourself by taking a Free Trial Session with us.

*No credit card is required, nor are you under any obligation to make a purchase. Just schedule the FREE TRIAL lesson to meet a tutor & get help on any topic you want!*

Our Learning by Design methodology focuses exclusively on individual students.

Our expert tutors are specially trained to identify and diagnose the needs and skills of each student and plan future tutoring lessons accordingly.

Know more about our Personalised Online Tutoring Packs.

+1-269-763-4602

+1-269-763-5024

#### Quick Links

#### Follow Us

#### Give Your Child The eTutorWorld Advantage

Research has proven that personal online tutoring not just cements school learning, it helps build student confidence. eTutorWorld provides the best K-12 Online Tutoring Services so you can learn from the comfort and safety of your home at an affordable cost.

Be it an exam, class test or a quiz, eTutorWorld’s Math, Science and English tutors are responsible for your academic progress. Meet your personal coach at your convenient day and time to get help for Grade 3-12 Math, Science and English subjects and AP, SAT, SSAT and SCAT Test Prep help and test practice. All our tutors are graduates and bring with them years of teaching experience to the tutoring lessons.

Our ‘*Learning by Design*’ methodology makes sure that each student is at the center of the teaching-learning process. All tutoring sessions start with a question to the student ‘*What do you want to learn today?*’ Hence, tutors diagnose your skills and recognize your requirements before the actual tutoring happens. Post every tutoring session, an individualized worksheet is emailed to the student to assimilate learned concepts. Regular formative assessments are used to evaluate a student’s understanding of the subject.

The state of the art technology used is stable, user friendly and safe. All you need is a computer or a tablet and an internet connection. The easy-to-use web conferencing software requires a one time download, using which the student can talk and chat to the tutor, annotate on an interactive shared whiteboard or even share documents, assignments or worksheets.

All tutoring sessions are recorded and made available for a month so you can review concepts taught.

Email or call our support team with any issues or questions – we are here for you 24X7.

Also, download free printable math and science worksheets in pdf format and solve SCAT and SSAT Practice Tests online. Sign up for a Free Trial Lesson Today!

Thousands have taken the eTutorWorld Advantage – what are you waiting for?

© 2020 eTutorWorld - Online Tutoring and Test Prep | All rights reserved