<< Chapter < Page Chapter >> Page >

Given two sides and the angle between them (SAS), find the measures of the remaining side and angles of a triangle.

  1. Sketch the triangle. Identify the measures of the known sides and angles. Use variables to represent the measures of the unknown sides and angles.
  2. Apply the Law of Cosines to find the length of the unknown side or angle.
  3. Apply the Law of Sines    or Cosines to find the measure of a second angle.
  4. Compute the measure of the remaining angle.

Finding the unknown side and angles of a sas triangle

Find the unknown side and angles of the triangle in [link] .

A triangle with standard labels. Side a = 10, side c = 12, and angle beta = 30 degrees.

First, make note of what is given: two sides and the angle between them. This arrangement is classified as SAS and supplies the data needed to apply the Law of Cosines.

Each one of the three laws of cosines begins with the square of an unknown side opposite a known angle. For this example, the first side to solve for is side b , as we know the measurement of the opposite angle β .

b 2 = a 2 + c 2 2 a c cos β b 2 = 10 2 + 12 2 2 ( 10 ) ( 12 ) cos ( 30 ) Substitute the measurements for the known quantities . b 2 = 100 + 144 240 ( 3 2 ) Evaluate the cosine and begin to simplify . b 2 = 244 120 3 b = 244 120 3 Use the square root property . b 6.013

Because we are solving for a length, we use only the positive square root. Now that we know the length b , we can use the Law of Sines to fill in the remaining angles of the triangle. Solving for angle α , we have

sin α a = sin β b sin α 10 = sin ( 30° ) 6.013 sin α = 10 sin ( 30° ) 6.013 Multiply both sides of the equation by 10 . α = sin 1 ( 10 sin ( 30° ) 6.013 ) Find the inverse sine of  10 sin ( 30° ) 6.013 . α 56.3°

The other possibility for α would be α = 180° 56.3° 123.7°. In the original diagram, α is adjacent to the longest side, so α is an acute angle and, therefore, 123.7° does not make sense. Notice that if we choose to apply the Law of Cosines    , we arrive at a unique answer. We do not have to consider the other possibilities, as cosine is unique for angles between and 180°. Proceeding with α 56.3° , we can then find the third angle of the triangle.

γ = 180° 30° 56.3° 93.7°

The complete set of angles and sides is

α 56.3° a = 10 β = 30° b 6.013 γ 93.7° c = 12

Find the missing side and angles of the given triangle: α = 30° , b = 12 , c = 24.

a 14.9 , β 23.8° , γ 126.2° .

Solving for an angle of a sss triangle

Find the angle α for the given triangle if side a = 20 , side b = 25 , and side c = 18.

For this example, we have no angles. We can solve for any angle using the Law of Cosines. To solve for angle α , we have

               a 2 = b 2 + c 2 −2 b c cos α               20 2 = 25 2 + 18 2 −2 ( 25 ) ( 18 ) cos α Substitute the appropriate measurements .               400 = 625 + 324 900 cos α Simplify in each step .               400 = 949 900 cos α            −549 = −900 cos α Isolate cos  α .            −549 −900 = cos α             0.61 cos α cos −1 ( 0.61 ) α Find the inverse cosine .                   α 52.4°

See [link] .

A triangle with standard labels. Side b =25, side a = 20, side c = 18, and angle alpha = 52.4 degrees.

Given a = 5 , b = 7 , and c = 10 , find the missing angles.

α 27.7° , β 40.5° , γ 111.8°

Solving applied problems using the law of cosines

Just as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is applicable to situations in which the given data fits the cosine models. We may see these in the fields of navigation, surveying, astronomy, and geometry, just to name a few.

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Contemporary math applications' conversation and receive update notifications?

Ask