Answer: 56 degrees
Step-by-step explanation:
We know that angle between b and c is 90 degrees. Because there is a line dividing angles a and b from 124 and c, we also know that each side is 180 degrees (a and b add to 180, and 124 and c add to 180).
124 and c are supplementary angles. We can represent this in an equation to solve for c:
![124 + c = 180\\\\c = 56](https://tex.z-dn.net/?f=124%20%2B%20c%20%3D%20180%5C%5C%5C%5Cc%20%3D%2056)
9t-6-6t=6 -- add 6 to both sides
9t-6t=12 -- combine like terms
3t=12 -- divide by 3
t=4
Answer: The process of combining matrices, vectors, or other quantities under specific rules to obtain their product.
Given:
![\sin (u)=-\dfrac{7}{25}](https://tex.z-dn.net/?f=%5Csin%20%28u%29%3D-%5Cdfrac%7B7%7D%7B25%7D)
![\cos (v)=-\dfrac{4}{5}](https://tex.z-dn.net/?f=%5Ccos%20%28v%29%3D-%5Cdfrac%7B4%7D%7B5%7D)
To find:
The exact value of cos(u-v) if both angles are in quadrant 3.
Solution:
In 3rd quadrant, cos and sin both trigonometric ratios are negative.
We have,
![\sin (u)=-\dfrac{7}{25}](https://tex.z-dn.net/?f=%5Csin%20%28u%29%3D-%5Cdfrac%7B7%7D%7B25%7D)
![\cos (v)=-\dfrac{4}{5}](https://tex.z-dn.net/?f=%5Ccos%20%28v%29%3D-%5Cdfrac%7B4%7D%7B5%7D)
Now,
![\cos (u)=-\sqrt{1-\sin^2 (u)}](https://tex.z-dn.net/?f=%5Ccos%20%28u%29%3D-%5Csqrt%7B1-%5Csin%5E2%20%28u%29%7D)
![\cos (u)=-\sqrt{1-(-\dfrac{7}{25})^2}](https://tex.z-dn.net/?f=%5Ccos%20%28u%29%3D-%5Csqrt%7B1-%28-%5Cdfrac%7B7%7D%7B25%7D%29%5E2%7D)
![\cos (u)=-\sqrt{1-\dfrac{49}{625}}](https://tex.z-dn.net/?f=%5Ccos%20%28u%29%3D-%5Csqrt%7B1-%5Cdfrac%7B49%7D%7B625%7D%7D)
![\cos (u)=-\sqrt{\dfrac{625-49}{625}}](https://tex.z-dn.net/?f=%5Ccos%20%28u%29%3D-%5Csqrt%7B%5Cdfrac%7B625-49%7D%7B625%7D%7D)
On further simplification, we get
![\cos (u)=-\sqrt{\dfrac{576}{625}}](https://tex.z-dn.net/?f=%5Ccos%20%28u%29%3D-%5Csqrt%7B%5Cdfrac%7B576%7D%7B625%7D%7D)
![\cos (u)=-\dfrac{24}{25}](https://tex.z-dn.net/?f=%5Ccos%20%28u%29%3D-%5Cdfrac%7B24%7D%7B25%7D)
Similarly,
![\sin (v)=-\sqrt{1-\cos^2 (v)}](https://tex.z-dn.net/?f=%5Csin%20%28v%29%3D-%5Csqrt%7B1-%5Ccos%5E2%20%28v%29%7D)
![\sin (v)=-\sqrt{1-(-\dfrac{4}{5})^2}](https://tex.z-dn.net/?f=%5Csin%20%28v%29%3D-%5Csqrt%7B1-%28-%5Cdfrac%7B4%7D%7B5%7D%29%5E2%7D)
![\sin (v)=-\sqrt{1-\dfrac{16}{25}}](https://tex.z-dn.net/?f=%5Csin%20%28v%29%3D-%5Csqrt%7B1-%5Cdfrac%7B16%7D%7B25%7D%7D)
![\sin (v)=-\sqrt{\dfrac{25-16}{25}}](https://tex.z-dn.net/?f=%5Csin%20%28v%29%3D-%5Csqrt%7B%5Cdfrac%7B25-16%7D%7B25%7D%7D)
![\sin (v)=-\sqrt{\dfrac{9}{25}}](https://tex.z-dn.net/?f=%5Csin%20%28v%29%3D-%5Csqrt%7B%5Cdfrac%7B9%7D%7B25%7D%7D)
![\sin (v)=-\dfrac{3}{5}](https://tex.z-dn.net/?f=%5Csin%20%28v%29%3D-%5Cdfrac%7B3%7D%7B5%7D)
Now,
![\cos (u-v)=\cos u\cos v+\sin u\sin v](https://tex.z-dn.net/?f=%5Ccos%20%28u-v%29%3D%5Ccos%20u%5Ccos%20v%2B%5Csin%20u%5Csin%20v)
![\cos (u-v)=\left(-\dfrac{24}{25}\right)\left(-\dfrac{4}{5}\right)+\left(-\dfrac{7}{25}\right)\left(-\dfrac{3}{25}\right)](https://tex.z-dn.net/?f=%5Ccos%20%28u-v%29%3D%5Cleft%28-%5Cdfrac%7B24%7D%7B25%7D%5Cright%29%5Cleft%28-%5Cdfrac%7B4%7D%7B5%7D%5Cright%29%2B%5Cleft%28-%5Cdfrac%7B7%7D%7B25%7D%5Cright%29%5Cleft%28-%5Cdfrac%7B3%7D%7B25%7D%5Cright%29)
![\cos (u-v)=\dfrac{96}{625}+\dfrac{21}{625}](https://tex.z-dn.net/?f=%5Ccos%20%28u-v%29%3D%5Cdfrac%7B96%7D%7B625%7D%2B%5Cdfrac%7B21%7D%7B625%7D)
![\cos (u-v)=\dfrac{1 17}{625}](https://tex.z-dn.net/?f=%5Ccos%20%28u-v%29%3D%5Cdfrac%7B1%2017%7D%7B625%7D)
Therefore, the value of cos (u-v) is 0.1872.
Answer:
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