Answer:
![x=(-1)^{k+1}\cdot \dfrac{\pi}{6}+\pi k,\ k\in Z](https://tex.z-dn.net/?f=x%3D%28-1%29%5E%7Bk%2B1%7D%5Ccdot%20%5Cdfrac%7B%5Cpi%7D%7B6%7D%2B%5Cpi%20k%2C%5C%20k%5Cin%20Z)
Step-by-step explanation:
Consider the equation
![2\sin^2x-5\sin x-3=0](https://tex.z-dn.net/?f=2%5Csin%5E2x-5%5Csin%20x-3%3D0)
Substitute
![u=\sin x](https://tex.z-dn.net/?f=u%3D%5Csin%20x)
into the equation:
![2u^2-5u-3=0](https://tex.z-dn.net/?f=2u%5E2-5u-3%3D0)
This is quadratic equation, where
![D=(-5)^2-4\cdot 2\cdot (-3)=25+24=49\\ \\\sqrt{D}=7](https://tex.z-dn.net/?f=D%3D%28-5%29%5E2-4%5Ccdot%202%5Ccdot%20%28-3%29%3D25%2B24%3D49%5C%5C%20%5C%5C%5Csqrt%7BD%7D%3D7)
Hence,
![u_1=\dfrac{-(-5)-\sqrt{D}}{2\cdot 2}=\dfrac{5-7}{4}=-\dfrac{1}{2}\\ \\u_2=\dfrac{-(-5)+\sqrt{D}}{2\cdot 2}=\dfrac{5+7}{4}=3](https://tex.z-dn.net/?f=u_1%3D%5Cdfrac%7B-%28-5%29-%5Csqrt%7BD%7D%7D%7B2%5Ccdot%202%7D%3D%5Cdfrac%7B5-7%7D%7B4%7D%3D-%5Cdfrac%7B1%7D%7B2%7D%5C%5C%20%5C%5Cu_2%3D%5Cdfrac%7B-%28-5%29%2B%5Csqrt%7BD%7D%7D%7B2%5Ccdot%202%7D%3D%5Cdfrac%7B5%2B7%7D%7B4%7D%3D3)
Now, use the substitution again:
1. When
then
![\sin x=-\dfrac{1}{2}\\ \\x=(-1)^k\sin^{-1}\left(-\dfrac{1}{2}\right)+\pi k,\ k\in Z\\ \\x=(-1)^k\cdot \left(-\dfrac{\pi}{6}\right)+\pi k,\ k\in Z\\ \\x=(-1)^{k+1}\cdot \dfrac{\pi}{6}+\pi k,\ k\in Z](https://tex.z-dn.net/?f=%5Csin%20x%3D-%5Cdfrac%7B1%7D%7B2%7D%5C%5C%20%5C%5Cx%3D%28-1%29%5Ek%5Csin%5E%7B-1%7D%5Cleft%28-%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%2B%5Cpi%20k%2C%5C%20k%5Cin%20Z%5C%5C%20%5C%5Cx%3D%28-1%29%5Ek%5Ccdot%20%5Cleft%28-%5Cdfrac%7B%5Cpi%7D%7B6%7D%5Cright%29%2B%5Cpi%20k%2C%5C%20k%5Cin%20Z%5C%5C%20%5C%5Cx%3D%28-1%29%5E%7Bk%2B1%7D%5Ccdot%20%5Cdfrac%7B%5Cpi%7D%7B6%7D%2B%5Cpi%20k%2C%5C%20k%5Cin%20Z)
2. When
then
![\sin x=3](https://tex.z-dn.net/?f=%5Csin%20x%3D3)
This equation has no solutions, because the range of
is
and 3>1.
160-1
157+1
158X1
158/1
2x79
Answer:
19a.) 6x-1=4 --> 6x=5 --> x=5/6
b.) 4w+3=10 --> 4w=7 --> w=7/4
Evaluate when t= -2
20a.) 5t-3 --> 5(-2)-3= -10-3= -13
b) the sequence 5+t, 5+2t, 5+3t, 5+4t. = 3, 1, -1, -3
c.) 18- t² =18-(-2)^2= 18-4=14
20d) the sequence 2t- 3, 2t-5, 2t-7, 2t-9. = -7, -9, -11, -13
21.) Find the value of these expressions when p=2, q=3 and r=6
21 a)5q = 5(3)=15
b) p+q
= 2+3= 5
c) qr
= 3*6= 18
d) 2p+ 3q
= 2(2)+3(3)=4+9= 13
e) r² = 6^2=36