Answer:
The measures of the angles at its corners are ![59.1\°,35.4\°,85.5\°](https://tex.z-dn.net/?f=59.1%5C%C2%B0%2C35.4%5C%C2%B0%2C85.5%5C%C2%B0)
Step-by-step explanation:
see the attached figure to better understand the problem
step 1
Find the measure of angle A
Applying the law of cosines
![185^{2}= 215^{2}+125^{2}-2(215)(125)cos(A)](https://tex.z-dn.net/?f=185%5E%7B2%7D%3D%20215%5E%7B2%7D%2B125%5E%7B2%7D-2%28215%29%28125%29cos%28A%29)
![2(215)(125)cos(A)= 215^{2}+125^{2}-185^{2}](https://tex.z-dn.net/?f=2%28215%29%28125%29cos%28A%29%3D%20215%5E%7B2%7D%2B125%5E%7B2%7D-185%5E%7B2%7D)
![cos(A)= [215^{2}+125^{2}-185^{2}]/(2(215)(125))](https://tex.z-dn.net/?f=cos%28A%29%3D%20%5B215%5E%7B2%7D%2B125%5E%7B2%7D-185%5E%7B2%7D%5D%2F%282%28215%29%28125%29%29)
![cos(A)=0.513953](https://tex.z-dn.net/?f=cos%28A%29%3D0.513953)
![A=arccos(0.513953)=59.1\°](https://tex.z-dn.net/?f=A%3Darccos%280.513953%29%3D59.1%5C%C2%B0)
step 2
Find the measure of angle B
Applying the law of cosines
![125^{2}= 215^{2}+185^{2}-2(215)(185)cos(B)](https://tex.z-dn.net/?f=125%5E%7B2%7D%3D%20215%5E%7B2%7D%2B185%5E%7B2%7D-2%28215%29%28185%29cos%28B%29)
![2(215)(185)cos(B)= 215^{2}+185^{2}-125^{2}](https://tex.z-dn.net/?f=2%28215%29%28185%29cos%28B%29%3D%20215%5E%7B2%7D%2B185%5E%7B2%7D-125%5E%7B2%7D)
![cos(B)= [215^{2}+185^{2}-125^{2}]/(2(215)(185))](https://tex.z-dn.net/?f=cos%28B%29%3D%20%5B215%5E%7B2%7D%2B185%5E%7B2%7D-125%5E%7B2%7D%5D%2F%282%28215%29%28185%29%29)
![cos(B)=0.81489](https://tex.z-dn.net/?f=cos%28B%29%3D0.81489)
![B=arccos(0.81489)=35.4\°](https://tex.z-dn.net/?f=B%3Darccos%280.81489%29%3D35.4%5C%C2%B0)
step 3
Find the measure of angle C
Applying the law of cosines
![215^{2}= 125^{2}+185^{2}-2(125)(185)cos(C)](https://tex.z-dn.net/?f=215%5E%7B2%7D%3D%20125%5E%7B2%7D%2B185%5E%7B2%7D-2%28125%29%28185%29cos%28C%29)
![2(125)(185)cos(C)= 125^{2}+185^{2}-215^{2}](https://tex.z-dn.net/?f=2%28125%29%28185%29cos%28C%29%3D%20125%5E%7B2%7D%2B185%5E%7B2%7D-215%5E%7B2%7D)
![cos(C)= [125^{2}+185^{2}-215^{2}]/(2(125)(185))](https://tex.z-dn.net/?f=cos%28C%29%3D%20%5B125%5E%7B2%7D%2B185%5E%7B2%7D-215%5E%7B2%7D%5D%2F%282%28125%29%28185%29%29)
![cos(C)=0.0784](https://tex.z-dn.net/?f=cos%28C%29%3D0.0784)
![C=arccos(0.0784)=85.5\°](https://tex.z-dn.net/?f=C%3Darccos%280.0784%29%3D85.5%5C%C2%B0)
Answer:
sin(theta) + cos(theta) = 0
sin(theta) = -cos(theta)
sin(theta)/cos(theta) = -1
tan(theta) = -1
theta = - 45° ± k·180°
Step-by-step explanation:
![x_{1,\:2}=\frac{-6\pm \sqrt{6^2-4\cdot \:1\cdot \left(-16\right)}}{2\cdot \:1}\\\sqrt{6^2-4\cdot \:1\cdot \left(-16\right)}\\\sqrt{6^2+64}\\\sqrt{36+64}\\\sqrt{100}\\\sqrt{10^2}\\=10\\x_1=\frac{-6+10}{2\cdot \:1},\:x_2=\frac{-6-10}{2\cdot \:1}\\\\\bold{x_1:2}\\\frac{-6+10}{2\cdot \:1}\\\frac{4}{2\cdot \:1}\\\frac{4}{2}\\=2\\\\\bold{x_2:-8}\\\frac{-6-10}{2\cdot \:1}\\\frac{-16}{2\cdot \:1}\\\frac{-16}{2}\\-\frac{16}{2}\\=-8](https://tex.z-dn.net/?f=x_%7B1%2C%5C%3A2%7D%3D%5Cfrac%7B-6%5Cpm%20%5Csqrt%7B6%5E2-4%5Ccdot%20%5C%3A1%5Ccdot%20%5Cleft%28-16%5Cright%29%7D%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5C%5Csqrt%7B6%5E2-4%5Ccdot%20%5C%3A1%5Ccdot%20%5Cleft%28-16%5Cright%29%7D%5C%5C%5Csqrt%7B6%5E2%2B64%7D%5C%5C%5Csqrt%7B36%2B64%7D%5C%5C%5Csqrt%7B100%7D%5C%5C%5Csqrt%7B10%5E2%7D%5C%5C%3D10%5C%5Cx_1%3D%5Cfrac%7B-6%2B10%7D%7B2%5Ccdot%20%5C%3A1%7D%2C%5C%3Ax_2%3D%5Cfrac%7B-6-10%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5C%5C%5C%5Cbold%7Bx_1%3A2%7D%5C%5C%5Cfrac%7B-6%2B10%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5C%5Cfrac%7B4%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5C%5Cfrac%7B4%7D%7B2%7D%5C%5C%3D2%5C%5C%5C%5C%5Cbold%7Bx_2%3A-8%7D%5C%5C%5Cfrac%7B-6-10%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5C%5Cfrac%7B-16%7D%7B2%5Ccdot%20%5C%3A1%7D%5C%5C%5Cfrac%7B-16%7D%7B2%7D%5C%5C-%5Cfrac%7B16%7D%7B2%7D%5C%5C%3D-8)
Answer:
x₁ = 2
x₂ = -8