# 10.1 Non-right triangles: law of sines  (Page 5/10)

 Page 5 / 10

## Verbal

Describe the altitude of a triangle.

The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.

Compare right triangles and oblique triangles.

When can you use the Law of Sines to find a missing angle?

When the known values are the side opposite the missing angle and another side and its opposite angle.

In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?

What type of triangle results in an ambiguous case?

A triangle with two given sides and a non-included angle.

## Algebraic

For the following exercises, assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha =43°,\gamma =69°,a=20$

$\alpha =35°,\gamma =73°,c=20$

$\alpha =60°,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =60°,\text{\hspace{0.17em}}\gamma =60°$

$a=4,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\text{\hspace{0.17em}}60°,\text{\hspace{0.17em}}\beta =100°$

$b=10,\text{\hspace{0.17em}}\beta =95°,\gamma =\text{\hspace{0.17em}}30°$

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ angle $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and angle $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.$

Find side $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}A=37°,\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=49°,\text{\hspace{0.17em}}c=5.$

$b\approx 3.78$

Find side $\text{\hspace{0.17em}}a$ when $\text{\hspace{0.17em}}A=132°,C=23°,b=10.$

Find side $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}B=37°,C=21,\text{\hspace{0.17em}}b=23.$

$c\approx 13.70$

For the following exercises, assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

$\alpha =119°,a=14,b=26$

$\gamma =113°,b=10,c=32$

one triangle, $\text{\hspace{0.17em}}\alpha \approx 50.3°,\beta \approx 16.7°,a\approx 26.7$

$b=3.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=5.3,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\gamma =\text{\hspace{0.17em}}80°$

$a=12,\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=17,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\text{\hspace{0.17em}}35°$

two triangles, or

$a=20.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=35.0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =25°$

$a=7,\text{\hspace{0.17em}}c=9,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\text{\hspace{0.17em}}43°$

two triangles, or

$a=7,b=3,\beta =24°$

$b=13,c=5,\gamma =\text{\hspace{0.17em}}10°$

two triangles, $\text{\hspace{0.17em}}\alpha \approx 143.2°,\beta \approx 26.8°,a\approx 17.3\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}{\alpha }^{\prime }\approx 16.8°,{\beta }^{\prime }\approx 153.2°,{a}^{\prime }\approx 8.3$

$a=2.3,c=1.8,\gamma =28°$

$\beta =119°,b=8.2,a=11.3$

no triangle possible

For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

Find angle $A$ when $\text{\hspace{0.17em}}a=24,b=5,B=22°.$

Find angle $A$ when $\text{\hspace{0.17em}}a=13,b=6,B=20°.$

$A\approx 47.8°\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}{A}^{\prime }\approx 132.2°$

Find angle $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}A=12°,a=2,b=9.$

For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

$a=5,c=6,\beta =\text{\hspace{0.17em}}35°$

$8.6$

$b=11,c=8,\alpha =28°$

$a=32,b=24,\gamma =75°$

$370.9$

$a=7.2,b=4.5,\gamma =43°$

## Graphical

For the following exercises, find the length of side $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Round to the nearest tenth.

$12.3$

For the following exercises, find the measure of angle $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ if possible. Round to the nearest tenth.

$29.7°$

Notice that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is an obtuse angle.

$110.6°$

For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.

$57.1$

## Extensions

Find the radius of the circle in [link] . Round to the nearest tenth.

Find the diameter of the circle in [link] . Round to the nearest tenth.

$10.1$

why {2kπ} union {kπ}={kπ}?
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
give me treganamentry question
Solve 2cos x + 3sin x = 0.5
madras university algebra questions papers first year B. SC. maths
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
y=x+1
gary
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Wilson
what is algebra
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace