<< 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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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° .

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

α 27.7° , β 40.5° , γ 111.8°

Got questions? Get instant answers now!

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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask