How do I solve these Trig Questions?

Question

1. What is the y-intercept of y = cos x?

2. Find the amplitude of y = -2cos (4x - pi).

Could someone please explain these? Thanks.

Answer 1

1) In cartesian coordinate systems, the y-intercept is the value of the function when x = 0. Thus ;

The Y intercept is Y = cos(0) = 1

The value of the Y-intercept is 1 and the 2D cartesian coordinate of the Y-intercept is (0,1)

(notice that the question seems to ask for the value of the Y-intercept, it doesn;t specify its coordinate)

2) The amplitude of a periodic function would be the maximum and minumum value of y for a complete cycle or period. For trig functions, the high and low points (or amplitude) will occur when the derivative of that function is 0.

Thus, step 1, we take the derivative of the function given ( y = -2cos(4x-pi) ) ;

dy/dx = -2 sin(4x - pi)

Set it to 0 and solve ;

-2sin(4x-pi) = 0 =>
sin(4x-pi) = 0 =>
4x - pi = asin(0) = 0 =>
4x - pi = 0
4x = pi
x = pi/4

So, the maximum amplitude will occur at x = pi/4

Thus, we calcaulate the amplitude at pi/4 (which is the value of Y from the original function ;

Y (amplitude) = -2 cos(4pi/4 - pi) = -2 cos( pi - pi) = -2cos(0)

Since cos(0) = 1

Amplitude = -2 * 1 = -2

But, relatively an amplitude is an absolute value, thus = 2

Answer 2

1 ) the y-intercept means the point where the graph intersects the y-axis. At this point, whatever the value of y is, you can be sure that the value of x is 0.

So you just plug in 0 for x in the equation
y = cos 0
y = 1
therefore the y-intercept is at (0,1)

2) the amplitude is something like the height of the graph, measured from the x=axis

if you sketch the graph of y = cos x, you will notice that the height is 1. This is supported by the fact that -1<=cosx,=1 for all values of x.

Now if you multiply cos x by -2, all values of y will become twice bigger. Therefore the amplitude is now 2, instead of 1.

The negativity of -2 will not affect the amplitude. It merely flips the graph about the x-axis.

The change from x to (4x-pi) in our argument will also not affect the amplitude. It merely moves the graph of y=cos x to the right, and increase the frequency by which the cycle repeats

Answer 3

1.) The point when x = 0 is where the curve crosses the y-axis (y-intercept), cos(0) = 1, therefore the point (0,1) is the y-intercept.

2.) The amplitude is 2. When you look at the function, (4x-pi) is what describes the length of the period of the wave and how it is shifted along the x-axis. The coefficient in front of the function is what describes its amplitude. The graph of 1*cos(x) has a maximum height of 1, so 2*cos(x) has a maximum height of 2. The negative sign only affects the point at which the wave reaches its maximum height. For example, consider the graphs of cos(x) and -cos(x), cos(0) = 1 and cos(pi) = -1, while -cos(0) = -1 and -cos(pi) = 1, however, both functions have the same amplitude.

Answer 4

The y intercept is the y value when x = 0 since the y axis crosses the x axis at x = 0. To solve for this you plug x = 0 into your eqaution of y = cos (x). Therefore, y = cos (0) for which cos(0) = 1. To answer your first question, the y-intercept of y = cos(x) is 1. To answer your second equation, all sin and cos functions are naturally 1 amplitude equations. To easily spot the amplitude of a sin or cos function, look at the coefficient of the equation. In your equation, the -2 is your coefficient. Therefore, your amplitude is -2 * 1 which is -2. Now, amplitude is an absolute value, thus, the amplitude is 2.

Answer 5

tan(3x)=?3 / 3 ?3 /3 is a undeniable ratio because of the fact the tangent of any 30°, or ?/6 radians, reference perspective would be in this ratio. on the x-y coordinate equipment, the ratio would be constructive purely in quadrants I and III, and detrimental in quadrants II and IV. interior the equation, ?3 / 3 is constructive, so evaluate all 30° reference angles in quadrants I and III. 3x=?/6+?N; N=any integer. (notice: ?/6+?=7?/6=210°, it rather is the perspective in quadrant III that has a 30° reference perspective. 7?/6+?=13?/6=390°, which has a reference perspective interior the quadrant I. consequently, each time ? is further to a reference perspective, the result would be yet another reference perspective.) x=?/18+?N/3; N=any integer.

Answer 6

Question 1
Y intercept occurs when x = 0
y = cos 0
y = 1 is y intercept.

Question 2
y = ± 2
Amplitude = 2

Answer 7

2. regular equation is y=a cos(t - k) + b
a= amplitude
t= period
k and b = phase shifts

so in this case... a = -2
but since amlitude doesnt have direction ( beside being in the y - axis)..

a= 2

Answer 8

1. x=0==>cos0=1==>the y-intercept of y = cos x is (0;1)

2. (no idea)

Answer 9

huh..