<< 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

what is the function of sine with respect of cosine , graphically
Karl Reply
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl 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