CPhill

Here's  (a)

Draw BV  and TC

In triangles TAC and BAV,  TA  = BA, AC  = AV

Angle TAB  = angle VAC  and angle BAC  = angle BAC

So....angle TAB  + angle BAC  = angle VAC  + angle BAC

So...angle TAC  = angle VAB

So by SAS, triangle TAC  is congruent to triangle BAV

So  TC  = BV

Now.....call the intersection of  BV and TC, N

And call the intersection of  AC and BV, O

Note that  angle TCA  = angle BVA

And angle AOV  = angle NOC

So.....by AA similarity, triangle AOV  is similar to triangle NOC

But angle OAV  is right.....so...angle ONC  is also right.....

Thus.....TC  is perpendicular to BV

Take care, Penquino !!!

-7pi/ 4  =  pi/4

So...the first answer is correct

cos ( - θ)  = cos ( θ )  = 4/5

Since  tan θ  and cos θ   >  0 , θ must be a first quadrant angle

So....sin θ  =  3/5

So.....sin (- θ)  = -sin (θ)  =  -3/5

1.  Here's the first one  : https://www.desmos.com/calculator/yxwbkrkywo

The midline  is  y  = -2

The first point is (pi/4, -2)

The second point  is  ( pi/2, -6)

2.  Here's the second one : https://www.desmos.com/calculator/a1sj0xcrjk

The midline is  y  = -5

The first point is (pi/4, - 5)

The second point is  (7pi/24, -8)

Positive  on   (-3, 1)   and  ( 4, infinity )

Notice that when x  = 0, we have that

f(0)  =  -2cos(4pi * 0 )  =   -2 cos (0)  =  - 2 (1)   = -2

So....notice that the second graph is the only  one  that contains the point  (0 , -2)

We can find the amplitude  by finding the difference between the max value on the graph and the min  value on the graph and dividing this difference by 2

Max  value   = 6

Min value  = -2

So

[ (6)  -  (-2)  ]  / 2   =   [ 8 ]  / 2   = 4  = the amplitude

6 / 15  =  2 / 5   = 40%     ⇒   Correct  !!!!

g.  The functions will return the same value when  x ≥ 0

      When  x  < 0,  the graph of g(x) will be the graph of  f(x)  reflected across the x axis

h  No such value exists.......when x ≤ 0,   the first function will produce a positive value  and the second function will produce the negative of this value....thus......their sum wil  = 0

i   Note that  when  x = 5 ..... f(x)  =  (5)^2 =  25   and  g (x)  = 5 l 5 l  = 25

So  f(5)  + g(5)  = 25  + 25   = 50