Question 1122860: Label each of the following statements as true or false.
The graph of y=cos (θ+ pi/2) is a reflection of the graph of y= -sin θ in the x-axis.
On the interval -pi < θ < pi, the only intersection point of the graphs of y= θ amd y= sin θ is at 0=0.
2. Given the right triangle ABC, fill the values of sin θ and cos θ, and prove that sin^2 θ + cos^2 θ= 1
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! The graph of y=cos (θ + pi/2) is a reflection of the graph of y= -sin θ in the x-axis.
i would say false.
these equations are identical.
if theta = pi/4 (45 degrees), then:
cos(pi/4 + pi/2) = -.7071067812
= sin(pi/4) = -.7071067812
this occurs at all values of theta.
the equations of y = cos(theta + pi/2) and y = sin(theta) would be reflections about the line = x.
the following graphs show the relationships.
in both graphs, i used x instead of theta, since x can be graphed easier than theta.
theta and x mean the same thing.
they are the angle being measured.
first graph is y = cos(x + pi/2) and y = -sin(x)
both equations give you the same graph which means the equations are equivalent.
second graph is y = cos(x + pi/2) and y = sin(x).
in this graph it is clear to see that the graph of y = sin(x) is a reflection of the graph y = cos(theta + pi/2) about the x-axis.
On the interval -pi < θ < pi, the only intersection point of the graphs of y= θ amd y= sin θ is at 0=0.
this statement is true as can be seen in the following graph.
in the graph, x represents theta.
they mean the same thing.
you can see that when x is 0, y = sin(x) is equal to 0 and y = x is also equal to 0.
at any other point on the graph, y = x is not equal to y = sin(x).
the graph would ahow all the intersection points between the two equations and only one is shown at x = 0.
 L
Given the right triangle ABC, fill the values of sin θ and cos θ, and prove that sin^2 θ + cos^2 θ= 1
in triangle ABC, side a is opposite angle A, side b is opposite angle B and side c is opposite angle C.
the hypotenuse of the triangle is the side opposite angle C which is the right angle in the triangle.
by pythagorus, c^2 = a^2 + b^2
that's a given.
you also know that sin(A) = opposite / hypotenuse = a/c and cos(A) = adjacent / hypotenuse = b/c
in sin(A) = a/c, you can solve for a to get a = c * sin(A).
in cos(A) = b/c, you can solve for b to get b = c * cos(A).
in the formula c^2 = a^2 + b^2, you can replace a with c * sin(A) and you can replace b with c * cos(A) to get:
c^2 = (c*sin(A))^2 + (c*cos(A))^2
this becomes:
c^2 = c^2 * sin^2(A) + c^2 * cos^2(A)
divide both sides of this equation by c^2 and you get:
1 = sin^2(A) + cos^2(A)
QED
the definition of QED is:
QED is an abbreviation of the Latin words "Quod Erat Demonstrandum" which loosely translated means "that which was to be demonstrated". It is usually placed at the end of a mathematical proof to indicate that the proof is complete.
|
|
|
