<< Chapter < Page Chapter >> Page >
Two oblique triangles with standard labels. Both have a dotted altitude line h extended from angle beta to the horizontal base side b. In the first, which is an acute triangle, the altitude is within the triangle. In the second, which is an obtuse triangle, the altitude h is outside of the triangle.

Thus,

Area = 1 2 ( base ) ( height ) = 1 2 b ( c sin α )

Similarly,

Area = 1 2 a ( b sin γ ) = 1 2 a ( c sin β )

Area of an oblique triangle

The formula for the area of an oblique triangle is given by

Area = 1 2 b c sin α = 1 2 a c sin β = 1 2 a b sin γ

This is equivalent to one-half of the product of two sides and the sine of their included angle.

Finding the area of an oblique triangle

Find the area of a triangle with sides a = 90 , b = 52 , and angle γ = 102° . Round the area to the nearest integer.

Using the formula, we have

Area = 1 2 a b sin γ Area = 1 2 ( 90 ) ( 52 ) sin ( 102° ) Area 2289 square units
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the area of the triangle given β = 42° , a = 7.2 ft , c = 3.4 ft . Round the area to the nearest tenth.

about 8.2 square feet

Got questions? Get instant answers now!

Solving applied problems using the law of sines

The more we study trigonometric applications, the more we discover that the applications are countless. Some are flat, diagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and motion.

Finding an altitude

Find the altitude of the aircraft in the problem introduced at the beginning of this section, shown in [link] . Round the altitude to the nearest tenth of a mile.

A diagram of a triangle where the vertices are the first ground station, the second ground station, and the airplane in the air between them. The angle between the first ground station and the plane is 15 degrees, and the angle between the second station and the airplane is 35 degrees. The side between the two stations is of length 20 miles. There is a dotted altitude line perpendicular to the ground side connecting the airplane vertex with the ground.

To find the elevation of the aircraft, we first find the distance from one station to the aircraft, such as the side a , and then use right triangle relationships to find the height of the aircraft, h .

Because the angles in the triangle add up to 180 degrees, the unknown angle must be 180°−15°−35°=130°. This angle is opposite the side of length 20, allowing us to set up a Law of Sines relationship.

   sin ( 130° ) 20 = sin ( 35° ) a a sin ( 130° ) = 20 sin ( 35° )                 a = 20 sin ( 35° ) sin ( 130° )                 a 14.98

The distance from one station to the aircraft is about 14.98 miles.

Now that we know a , we can use right triangle relationships to solve for h .

sin ( 15° ) = opposite hypotenuse sin ( 15° ) = h a sin ( 15° ) = h 14.98             h = 14.98 sin ( 15° )            h 3.88

The aircraft is at an altitude of approximately 3.9 miles.

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

The diagram shown in [link] represents the height of a blimp flying over a football stadium. Find the height of the blimp if the angle of elevation at the southern end zone, point A, is 70°, the angle of elevation from the northern end zone, point B , is 62°, and the distance between the viewing points of the two end zones is 145 yards.

An oblique triangle formed from three vertices A, B, and C. Verticies A and B are points on the ground, and vertex C is the blimp in the air between them. The distance between A and B is 145 yards. The angle at vertex A is 70 degrees, and the angle at vertex B is 62 degrees.

161.9 yd.

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with trigonometric applications.

Key equations

Law of Sines sin α a = sin β b = sin γ c a sin α = b sin β = c sin γ
Area for oblique triangles Area = 1 2 b c sin α         = 1 2 a c sin β         = 1 2 a b sin γ

Key concepts

  • The Law of Sines can be used to solve oblique triangles, which are non-right triangles.
  • According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side.
  • There are three possible cases: ASA, AAS, SSA. Depending on the information given, we can choose the appropriate equation to find the requested solution. See [link] .
  • The ambiguous case arises when an oblique triangle can have different outcomes.
  • There are three possible cases that arise from SSA arrangement—a single solution, two possible solutions, and no solution. See [link] and [link] .
  • The Law of Sines can be used to solve triangles with given criteria. See [link] .
  • The general area formula for triangles translates to oblique triangles by first finding the appropriate height value. See [link] .
  • There are many trigonometric applications. They can often be solved by first drawing a diagram of the given information and then using the appropriate equation. See [link] .

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask