The graph of sinx and cosx intersect once between 0 and pi/2. What is the angle between the two curves at the point where they intersect? (You need to think about how the angle between two curves should be defined).

first find their intersection

sinx = cosx
sinx/cosx = 1
tanx = 1
x = 45° or π/4 radians

for y = sinx , dy/dx = cosx
so at x = π/4 , dy/dx = 1/√2
tan^-1(1/√2) = 35.26°

for y = cosx , dy/dx = -sinx
so at x = π/4 , dy/dx = -1/√2
tan^-1(-1/√2) = 144.74°

angle between the two tangents = 144.74 - 35.26 = 109.48°

set your calculator to radians if you need your answer in radians.

To find the angle between the curves of sine (sinx) and cosine (cosx) at the point where they intersect, let's consider the slopes of the curves at that point.

The derivative of sinx is cosx, and the derivative of cosx is -sinx. At the point of intersection, the slopes of the two curves are equal, so we'll set their derivatives equal to each other:

cosx = -sinx

To solve this equation, divide both sides by cosx (assuming cosx is not zero):

1 = -tanx

Now take the arctan (inverse tangent) of both sides to find x:

x = arctan(-1)

Since we're given that the point of intersection is between 0 and π/2, the value of x is π/4 (45 degrees).

Now, to calculate the angle between the two curves at this point, we can find the difference between the derivative of sinx and the derivative of cosx:

cos(π/4) - (-sin(π/4)) = (√2)/2 + (√2)/2 = √2

Therefore, the angle between the two curves at the point of intersection is √2 radians or approximately 1.57 radians.

To find the angle between the two curves, we need to think about the concept of slope or gradient. The angle between two curves at a point of intersection is defined as the angle between their tangent lines at that point.

For the functions sin(x) and cos(x), the tangent lines are given by the derivatives of the functions at the point of intersection. Let's find the derivatives first:

The derivative of sin(x) is cos(x).
The derivative of cos(x) is -sin(x).

Since we know the two curves intersect once between 0 and pi/2, we need to find the x-coordinate of the point of intersection. To do this, we set sin(x) equal to cos(x) and solve for x:

sin(x) = cos(x)

Divide both sides by cos(x):

tan(x) = 1

Now, solving for x:

x = arctan(1)

By evaluating this expression, we find that x = pi/4.

Next, we find the slopes of the tangent lines at x = pi/4. We evaluate the derivatives of sin(x) and cos(x) at x = pi/4:

cos(pi/4) = sqrt(2)/2
-sin(pi/4) = -sqrt(2)/2

So, the slopes of the tangent lines are sqrt(2)/2 for sin(x) and -sqrt(2)/2 for cos(x) at x = pi/4.

Finally, to find the angle between the two curves at the point of intersection, we can use the formula for the angle between two lines:

angle = arctan((m2 - m1) / (1 + m1 * m2))

where m1 and m2 are the slopes of the two lines. Substituting the values, we get:

angle = arctan((-sqrt(2)/2 - sqrt(2)/2) / (1 + sqrt(2)/2 * sqrt(2)/2))
= arctan(-sqrt(2) / (1 + 1/2))
= arctan(-sqrt(2) / (3/2))
= arctan(-2sqrt(2) / 3)

Using a calculator to evaluate the arctan function, we find that the angle between the two curves at the point of intersection is approximately -56.31 degrees.