In the triangle shown on the right, ðp = 55°, n = 10, and g = 12. find the remaining side and the two angles. round all answers to the nearest tenth. length of side p =

p = 10.3

m∠N = 52.7°

m∠G = 72.3°

Step-by-step explanation:

In ΔPNG

∵ m∠P = 55°

∵ n = 10 and g = 12

By using cos Rule:

∵ p² = n² + g² - 2(n)(g)cosP

∴ p² = 100 + 144 - 2(10)(12)cos(55°) = 106.341655

∴ p = 10.3

By using sin Rule:

10/sin(N) = 12/sin(G) = 10.3/sin55

∴ sinN = 10(sin55)/10.3 = 0.7952932

∴ m∠N = 52.7°

∴ sinG = 12(sin55)/10.3 = 0.95435189

∴ m∠G = 72.3°

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

60: 45 =12: 9 =4: 3 so the answer is 4: 3