# Online Trigonometry Tutoring: Laws of sines & cosines

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## 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 3
^{rd}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 ****Ð***A ***= 45****°****. Find the value of ****Ð***B***. **

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

10/sin 45 = 5/sin *B*

sin B = 1/2√2 or Ö2/4 (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} = 52 + 102 – 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*.

**Ambiguous Case – Law of ****Sines**

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

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

**Ambiguous Case – Law of ****Sines**

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

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

**Ambiguous Case – Law of ****Sines**

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

**Ambiguous Case – Law of ****Sines**

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

**Ambiguous Case – Law of ****Sines**

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

**Ambiguous Case – Law of ****Sines**** Example**

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

**Ambiguous Case – Law of ****Sines**** Example (cont…)**

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**

**Answers**

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

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