<< 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

(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
modul questions trigonometry
Thamarai Reply
(1+cosA)(1-cosA)=sin^2A
BINCY
differentiate f(t)=1/4t to the power 4 +8
Jessica Reply
I need trigonometry,polynomial
duru Reply
ok
Augustine
Why is 7 on top
Bertha Reply
simplify cot x / csc x
Catherine Reply
👉🌹Solve🌻 Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
what is the period of cos?
SIYAMTHEMBA Reply
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
Patrick
simplify: cot x/csc x
Catherine
sorry i didnt realize you were actually asking someone else to put their question on here. i thought this was where i was supposed to.
Catherine
some to dereve formula for bulky density
kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
questions
Thamarai
ok
AjA
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah 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