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a=7.8... <C=84.8°... <B=44.19

no solution

a=9... <B=58.2°... <C=82.4°

<A=112.2°... <B=28.8°... <C=39°

solve the triangle using the law of cosines if (round ANSWERS…

How many triangles, if any, will result from this set of data…

solve the triangle using the law of cosines if (round ANSWERS…

solve the triangle if (round ANSWERS to the nearest tenth) :…

a=7.8... <C=84.8°... <B=44.19

solve the triangle using the law of cosines if (round ANSWERS…

no solution

How many triangles, if any, will result from this set of data…

find the area of ∆DEF, if (round answe…

solve the triangle using the law of co…

How many triangles, if any, will resul…

solve the triangle using the law of co…

63.9 units²

a=7.8... <C=84.8°... <B=44.19

no solution

a=9... <B=58.2°... <C=82.4°

find the area of ∆DEF, if (round answe…

63.9 units²

solve the triangle using the law of co…

a=7.8... <C=84.8°... <B=44.19

63.9 units²

a=7.8... <C=84.8°... <B=44.19

no solution

a=9... <B=58.2°... <C=82.4°

find the area of ∆DEF, if (round answer to the nearest tenth)…

solve the triangle using the law of cosines if (round ANSWERS…

How many triangles, if any, will result from this set of data…

solve the triangle using the law of cosines if (round ANSWERS…

63.9 units²

find the area of ∆DEF, if (round answer to the nearest tenth)…

a=7.8... <C=84.8°... <B=44.19

solve the triangle using the law of cosines if (round ANSWERS…

solve the triangle using the law of co…

solve the triangle using the law of co…

solve the triangle if (round ANSWERS t…

find side a if (hint* use law of sines…

a=7.8... <C=84.8°... <B=44.19

a=9... <B=58.2°... <C=82.4°

<A=112.2°... <B=28.8°... <C=39°

a=10

solve the triangle using the law of co…

a=7.8... <C=84.8°... <B=44.19

solve the triangle using the law of co…

a=9... <B=58.2°... <C=82.4°

find the area of ∆DEF, if (round answe…

solve the triangle using the law of co…

How many triangles, if any, will resul…

solve the triangle using the law of co…

63.9 units²

a=7.8... <C=84.8°... <B=44.19

no solution

a=9... <B=58.2°... <C=82.4°

find the area of ∆DEF, if (round answe…

63.9 units²

solve the triangle using the law of co…

a=7.8... <C=84.8°... <B=44.19

find the area of ∆DEF, if (round answe…

solve the triangle using the law of co…

How many triangles, if any, will resul…

solve the triangle using the law of co…

63.9 units²

a=7.8... <C=84.8°... <B=44.19

no solution

a=9... <B=58.2°... <C=82.4°

find the area of ∆DEF, if (round answe…

63.9 units²

solve the triangle using the law of co…

a=7.8... <C=84.8°... <B=44.19

Law of Cosines: case 1

Law of Cosines: case 2

Law of Sines: case 3

Law of Sines: case 4

Use this in a non-right triangle when you know all three side…

Use this in a non-right triangle when you know two sides and…

Use this in a non-right triangle when you know two sides and…

Use this in a non-right triangle when you know two angles and…

Law of Cosines: case 1

Use this in a non-right triangle when you know all three side…

Law of Cosines: case 2

Use this in a non-right triangle when you know two sides and…

solve the triangle using the law of co…

solve the triangle using the law of co…

solve the triangle if (round ANSWERS t…

find side a if (hint* use law of sines…

a=7.8... <C=84.8°... <B=44.19

a=9... <B=58.2°... <C=82.4°

<A=112.2°... <B=28.8°... <C=39°

a=10

solve the triangle using the law of co…

a=7.8... <C=84.8°... <B=44.19

solve the triangle using the law of co…

a=9... <B=58.2°... <C=82.4°

what is the law of sines?

what is the law of sines used for?

how do you find the area using the law…

how do you find the perimeter?

sinA/a= sinB/b= sinC/c ... a/sinA = b/sinB = c/sinC

non right angle triangles

.5(side)(side)(sin*included angle) ... area= 1/2absinC... area= 1/…

add up all three sides

what is the law of sines?

sinA/a= sinB/b= sinC/c ... a/sinA = b/sinB = c/sinC

what is the law of sines used for?

non right angle triangles

Law of sines (AAS, ASA)

Law of cosines (SSS, SAS) 1

Law of cosines (SSS, SAS) 2

Law of cosines (SSS, SAS) 3

(a/sinA) = (b/sinB) = (c/sinC)

c^2 = a^2 + b^2 - 2abcosC

b^2 = a^2 + c^2 - 2accosB

a^2 = b^2 + c^2 - 2bccosA

Law of sines (AAS, ASA)

(a/sinA) = (b/sinB) = (c/sinC)

Law of cosines (SSS, SAS) 1

c^2 = a^2 + b^2 - 2abcosC

you use this law when given a triangle…

you use this law when given a triangle…

Law of Sines: What is the formula to f…

Law of Sines: What is the formula to f…

law of sines

law of cosines

sinA/a = sinB/b = sinC/c

a/sinA = b/sinB = c/sinC

you use this law when given a triangle…

law of sines

you use this law when given a triangle…

law of cosines

Law of Sines

Law of Cosines

Law of Cosines Rule of Thumb with SAS

Law of Cosines Rule of Thumb with SSS

a/sinA = b/sinB... AAS... ASA... SSA (many options)

SAS... SSS... a^2=b^2+c^2-2bc(cosA)

Find smallest angle next

Find the largest first

Law of Sines

a/sinA = b/sinB... AAS... ASA... SSA (many options)

Law of Cosines

SAS... SSS... a^2=b^2+c^2-2bc(cosA)

Law of Sines Formula (both ways)

Area of an Oblique Triangle (SAS)

How to find the height:

Give the conditions for when angle A i…

sinA/a = sinB/b = sinC/c OR a/sinA = b/sinB = c/sinC

A = 1/2bcSinA OR 1/2acSinB OR 1/2abSinC

h = bsinA

a ≤ b - no solution ... a > b - 1 solution

Law of Sines Formula (both ways)

sinA/a = sinB/b = sinC/c OR a/sinA = b/sinB = c/sinC

Area of an Oblique Triangle (SAS)

A = 1/2bcSinA OR 1/2acSinB OR 1/2abSinC

Law of sines

Law of sines: A,B,C are

Law of sines: a,b,c are

Law of sines

Sin A Sin B Sin C... _______= ______=_______... a .......... b ...…

Angles

Opposite angles of same name

Any time you know the measure of one angle and the length of…

Law of sines

Sin A Sin B Sin C... _______= ______=_______... a .......... b ...…

Law of sines: A,B,C are

Angles

Law of Sines

Area of an Oblique Triangle

Law of Cosines (finding a missing side…

Law of Cosines (finding a missing angle)

a/sinA = b/sinB = c/sinC

A=(1/2)bc(sinA)

a^2 = b^2 +c^2 -2bc(cosA)

cosA = (b^2 + c^2 - a^2)/(2bc)

Law of Sines

a/sinA = b/sinB = c/sinC

Area of an Oblique Triangle

A=(1/2)bc(sinA)

Law of Sines

Ambiguous Case (A is acute)

Ambiguous Case (A is obtuse)

Area law of Sines

ASA... AAS... SSA *Ambiguous case

a<h, no solution... a=h, one triangle (right)... a≥b, one triangle…

a≤b, no solution... a>b, one triangle

1/2(b)(h)... 1/2ac(sin B)... 1/2ab(sin C)... 1/2bc(sin A)

Law of Sines

ASA... AAS... SSA *Ambiguous case

Ambiguous Case (A is acute)

a<h, no solution... a=h, one triangle (right)... a≥b, one triangle…

Law of Sines (option 1)

Law of Sines (option 2)

Ambiguous Triangle

Law of Cosines (A)

sinA/a = sinB/b = sinC/c

a/sinA = b/sinB = c/sinC

when we are given a single angle. (SSA---sin)... 1. No triangle…

a2 = b2 + c2 - 2bccos A

Law of Sines (option 1)

sinA/a = sinB/b = sinC/c

Law of Sines (option 2)

a/sinA = b/sinB = c/sinC

Law of Sines

Law of Cosines

For SAS triangles using law of cosines

For SSS triangles using law of cosines

sinA/a=sinB/b=sinC/c

a^2=b^2 + C^2 -2bc(cosA)

find side first, then solve for the smallest angle

Solve for the largest angle first, then next largest

Law of Sines

sinA/a=sinB/b=sinC/c

Law of Cosines

a^2=b^2 + C^2 -2bc(cosA)

What is the formula for the law of sin…

What is the formula for the law of cos…

When does the law of sines have partic…

In the case of an ASS triangle, how do…

side a/sinA = side b/sinB

a²=b²+c²-2×b×c×cos(A)

When it's an ASS triangle

If sin(x) > 1

What is the formula for the law of sin…

side a/sinA = side b/sinB

What is the formula for the law of cos…

a²=b²+c²-2×b×c×cos(A)

Law of cosines equation

Law of cosines re-arranged to find ang…

How many sides and angles must be pres…

Law of sines equation

c²=a²+b²-2abcos(C)

cos(C)= a²+b²-c²÷2ab

two

a/sinA=b/sinB=c/sinC

Law of cosines equation

c²=a²+b²-2abcos(C)

Law of cosines re-arranged to find ang…

cos(C)= a²+b²-c²÷2ab

1/2absinC= 1/2bcsinA = 1/2acsinB

a/sinA = b/sinB = c/sinC

AAS, ASA, SSA

ASS

area of oblique triangles (law of sines)

law of sines

law of sines is used with what types of triangles: ____ _____…

ambiguous case:

1/2absinC= 1/2bcsinA = 1/2acsinB

area of oblique triangles (law of sines)

a/sinA = b/sinB = c/sinC

law of sines

oblique triangle

Law of sines is used to solve...

Law of sines

what letters mean what in labeling a t…

a triangle that does not have a right angle

ASA, AAS, SSA

a/sinA = b/sinB = c/sinC

lowercase means side and capital is an angle

oblique triangle

a triangle that does not have a right angle

Law of sines is used to solve...

ASA, AAS, SSA

Law of Sines

Area of a Triangle

Apothem

Law of Cosines

In any triangle, the ration between the sine of any angle and…

To find the area of any triangle, we take half the product of…

segment from the center of the polygon perpendicular to the s…

a^2 = b^2 + c^2 - 2bc COSA

Law of Sines

In any triangle, the ration between the sine of any angle and…

Area of a Triangle

To find the area of any triangle, we take half the product of…

Given SSS or SAS

After using LOC, solve for ______ with…

Area of triangle

Law of sines

Law of cosines... a^2 = b^2+c^2-2bccosA ... b^2 = a^2+c^2-2accosB…

Smallest angle

1/2•base•height

sin A/a =sin B/b = sin C/c

Given SSS or SAS

Law of cosines... a^2 = b^2+c^2-2bccosA ... b^2 = a^2+c^2-2accosB…

After using LOC, solve for ______ with…

Smallest angle