<< Chapter < Page Chapter >> Page >

Given α = 80° , a = 120 , and b = 121 , find the missing side and angles. If there is more than one possible solution, show both.

Solution 1

α = 80° a = 120 β 83.2° b = 121 γ 16.8° c 35.2

Solution 2

α = 80° a = 120 β 96.8° b = 121 γ 3.2° c 6.8
Got questions? Get instant answers now!

Solving for the unknown sides and angles of a ssa triangle

In the triangle shown in [link] , solve for the unknown side and angles. Round your answers to the nearest tenth.

An oblique triangle with standard labels. Side b is 9, side c is 12, and angle gamma is 85. Angle alpha, angle beta, and side a are unknown.

In choosing the pair of ratios from the Law of Sines to use, look at the information given. In this case, we know the angle γ = 85° , and its corresponding side c = 12 , and we know side b = 9. We will use this proportion to solve for β .

sin ( 85° ) 12 = sin β 9 Isolate the unknown . 9 sin ( 85° ) 12 = sin β

To find β , apply the inverse sine function. The inverse sine will produce a single result, but keep in mind that there may be two values for β . It is important to verify the result, as there may be two viable solutions, only one solution (the usual case), or no solutions.

β = sin 1 ( 9 sin ( 85° ) 12 ) β sin 1 ( 0.7471 ) β 48.3°

In this case, if we subtract β from 180°, we find that there may be a second possible solution. Thus, β = 180° 48.3° 131.7° . To check the solution, subtract both angles, 131.7° and 85°, from 180°. This gives

α = 180° 85° 131.7° 36.7° ,

which is impossible, and so β 48.3° .

To find the remaining missing values, we calculate α = 180° 85° 48.3° 46.7° . Now, only side a is needed. Use the Law of Sines to solve for a by one of the proportions.

  sin ( 85 ° ) 12 = sin ( 46.7 ° ) a a sin ( 85 ° ) 12 = sin ( 46.7 ° )             a = 12 sin ( 46.7 ° ) sin ( 85 ° ) 8.8

The complete set of solutions for the given triangle is

α 46.7°         a 8.8 β 48.3°         b = 9 γ = 85°             c = 12
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given α = 80° , a = 100 , b = 10 , find the missing side and angles. If there is more than one possible solution, show both. Round your answers to the nearest tenth.

β 5.7° , γ 94.3° , c 101.3

Got questions? Get instant answers now!

Finding the triangles that meet the given criteria

Find all possible triangles if one side has length 4 opposite an angle of 50°, and a second side has length 10.

Using the given information, we can solve for the angle opposite the side of length 10. See [link] .

sin α 10 = sin ( 50° ) 4 sin α = 10 sin ( 50° ) 4 sin α 1.915
An incomplete triangle. One side has length 4 opposite a 50 degree angle, and a second side has length 10 opposite angle a. The side of length 4 is too short to reach the side of length 10, so there is no third angle.

We can stop here without finding the value of α . Because the range of the sine function is [ 1 , 1 ] , it is impossible for the sine value to be 1.915. In fact, inputting sin 1 ( 1.915 ) in a graphing calculator generates an ERROR DOMAIN. Therefore, no triangles can be drawn with the provided dimensions.

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

Determine the number of triangles possible given a = 31 , b = 26 , β = 48° .

two

Got questions? Get instant answers now!

Finding the area of an oblique triangle using the sine function

Now that we can solve a triangle for missing values, we can use some of those values and the sine function to find the area of an oblique triangle. Recall that the area formula for a triangle is given as Area = 1 2 b h , where b is base and h is height. For oblique triangles, we must find h before we can use the area formula. Observing the two triangles in [link] , one acute and one obtuse, we can drop a perpendicular to represent the height and then apply the trigonometric property sin α = opposite hypotenuse to write an equation for area in oblique triangles. In the acute triangle, we have sin α = h c or c sin α = h . However, in the obtuse triangle, we drop the perpendicular outside the triangle and extend the base b to form a right triangle. The angle used in calculation is α , or 180 α .

Questions & Answers

prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
can't prove
Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
Akugry
hi
Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
Practice Key Terms 4

Get the best Algebra and trigonometry course in your pocket!





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