Using cos addition formula:
use x for theta
cos(x+π/6)=cosx*cos(π/6)-sinx*sin(π/6)
sinx=1/4
cosx=√15/4
cos(π/6)=√3/2
sin(π/6)=1/2
cos(x+π/6)=(√15/4*√3/2)-(1/4*1/2)
cos(x+π/6)=(√45/8)-(1/8 )
cos(x+π/6)=(√45-1)/8)
Answer:
- vertical scaling by a factor of -4
- horizontal translation 5 units left
- vertical translation 11 units up
Step-by-step explanation:
We notice that the multiplier of the squared term in f(x) is 0.5; in g(x), it is -2, so is a factor of -4 times that in f(x).
If we scale f(x) by a factor of -4, we get ...
-4f(x) = -2(x -2)² -12
In order for the squared quantity to be x+3, we have to add 5 to the value that is squared in f(x). That is, x -2 must become x +3. We have to replace x with (x+5) to do that, so ...
(x+5) -2 = x +3
The replacement of x with x+5 amounts to a translation of 5 units to the left.
We note that the added constant after our scaling changes from +3 to -12. Instead, we want it to be -1, so we must add 11 to the scaled function. That translates it upward by 11 units.
The attached graph shows the scaled and translated function g(x):
g(x) = -4f(x +5) +11
Answer: b = 17
Step-by-step explanation: The figure shows a triangle which upon closer observation is actually two triangles placed one inside the other. We have triangle QTR (the larger one) and triangle PTS (the smaller one).
The line PS is parallel to line QR, so in effect what we have here are two similar triangles. The ratios of similarity can be derived as
Line QT/line TR = line PT/line TS OR
Line QT/line QR = line PT/line PS.
With these ratios in mind we can now write the following expressions from the similar triangles;
QT = 2b + (2b - 17) and
TR = 16 + 8
TR = 24
Hence,
2b/16 = 2b + (2b - 17)/24
(That is, PT/TS = QT/TR)
2b/16 = (4b - 17)/24
By cross multiplication we now have,
2b(24) = 16(4b - 17)
By expanding the brackets we now have
48b = 64b - 272
By collecting like terms, 64b now moves to the left side of the equation and becomes negative
48b - 64b = -272
-16b = -272
Divide both sides of the equation by -16
b = 17
**Note** A negative number divided by another negative number yields a positive answer.
