Answer:
7. 49
8. 10.4
Step-by-step explanation:
7) our hypotenuse is 8, and the side opposite the base of ladder and ground angle is 6, therefore we can use Sin
sin = opposite/hypotenuse
sin(x) = 6/8 = 3/4
sin-1(3/4) = 49
8) using the theorem that alternate exterior angles are congruent
the angle opposite side x is 30
tan = opposite/adjacent
tan(30) = x/18
x = 18tan(30) = 6√(3) = 10.4
you can also use 30-60-90 special right triangles.
the side opposite 30 is x
side opposite 60 is x√(3)
and side opposite 90 is 2x
Answer:
60
Step-by-step explanation:
Since equalateral. The angles are the same.
First of all, you can use
![\sin^2(x)+\cos^2(x)=1 \iff \sin^2(x)=1-\cos^2(x)](https://tex.z-dn.net/?f=%5Csin%5E2%28x%29%2B%5Ccos%5E2%28x%29%3D1%20%5Ciff%20%5Csin%5E2%28x%29%3D1-%5Ccos%5E2%28x%29)
and the equation becomes
![1-\cos^2(x)=\cos^2\left(\dfrac{x}{2}\right)](https://tex.z-dn.net/?f=1-%5Ccos%5E2%28x%29%3D%5Ccos%5E2%5Cleft%28%5Cdfrac%7Bx%7D%7B2%7D%5Cright%29)
Now, from the known identity
![\cos^2(x)=\dfrac{1+\cos(2x)}{2}](https://tex.z-dn.net/?f=%5Ccos%5E2%28x%29%3D%5Cdfrac%7B1%2B%5Ccos%282x%29%7D%7B2%7D)
we can half all the angles and we get
![\cos^2\left(\dfrac{x}{2}\right)=\dfrac{1+\cos(x)}{2}](https://tex.z-dn.net/?f=%5Ccos%5E2%5Cleft%28%5Cdfrac%7Bx%7D%7B2%7D%5Cright%29%3D%5Cdfrac%7B1%2B%5Ccos%28x%29%7D%7B2%7D)
So, the equation has become
![1-\cos^2(x)=\dfrac{1+\cos(x)}{2} \iff 2-2\cos^2(x)=1+\cos(x)](https://tex.z-dn.net/?f=1-%5Ccos%5E2%28x%29%3D%5Cdfrac%7B1%2B%5Ccos%28x%29%7D%7B2%7D%20%5Ciff%202-2%5Ccos%5E2%28x%29%3D1%2B%5Ccos%28x%29)
So, everything comes down to solve
![2\cos^2(x)+\cos(x)-1=0](https://tex.z-dn.net/?f=2%5Ccos%5E2%28x%29%2B%5Ccos%28x%29-1%3D0)
The associated equation
![2t^2+t-1=0](https://tex.z-dn.net/?f=2t%5E2%2Bt-1%3D0)
has roots
![t=-1,\quad t=\dfrac{1}{2}](https://tex.z-dn.net/?f=t%3D-1%2C%5Cquad%20t%3D%5Cdfrac%7B1%7D%7B2%7D)
So, we want one of the following
![\cos(x)=-1,\quad \cos(x)=\dfrac{1}{2}](https://tex.z-dn.net/?f=%5Ccos%28x%29%3D-1%2C%5Cquad%20%5Ccos%28x%29%3D%5Cdfrac%7B1%7D%7B2%7D)
Solve for the associated angles and you're done
Hope this helps! This is steps on how to complete it & answer at bottom