A pendulum of length 10 cm has swung so that theta is the radian measure of the angle formed by the pendulum and a vertical line .Show that the number of centimeters in the vertical height of the end of the pendulum above its lowest position is 20 sin squared 1/2 theta...

h = L (1-cos T) geometry exact

cos 2A = 1 - 2 sin^2 A identity
so
cos T = 1 - 2 sin^2 (T/2)

h = L [ 1 - (1 -2 sin^2 (T/2) ]

= L [2 sin^2 (T/2) ]

= 20 sin^2 (T/2)

A pendulum of length 10 cm has swung so that 𝜃 is the radian measure of the angle formed by the

pendulum and a vertical line. If h(𝜃) centimeters is the vertical height of the end of the pendulum
above its lowest position, find the instantaneous rate of change of h(𝜃) with respect to 𝜃 when
𝜃 =
1
6
𝜋

To find the vertical height of the end of the pendulum above its lowest position, we can use trigonometric principles.

Let's break down the problem step-by-step:

Step 1: Draw the vertical line and the pendulum in its initial position.

Step 2: Let's denote the vertical height of the end of the pendulum above its lowest position as "h."

Step 3: We can observe that the angle formed by the pendulum and the vertical line is equal to theta.

Step 4: Using trigonometric principles, we can express h as the vertical component of the pendulum's movement: h = l * sin(theta), where l is the length of the pendulum (in this case, 10 cm).

Step 5: Substitute l = 10 cm into the above equation: h = 10 * sin(theta).

Step 6: Now, we need to simplify the expression. Recall the double angle identity for sine: sin^2(theta/2) = (1 - cos(theta)) / 2.

Step 7: Divide both sides of the equation by 2, and multiply by 20 to obtain: 20 * sin^2(theta/2) = 10 * (1 - cos(theta)).

Step 8: Since cos(theta) can be expressed as 1 - 2sin^2(theta/2), we substitute this value into the equation: 20 * sin^2(theta/2) = 10 * (1 - (1- 2sin^2(theta/2))).

Step 9: Simplify the equation: 20 * sin^2(theta/2) = 10 * (2sin^2(theta/2)).

Step 10: Simplify further: 20 * sin^2(theta/2) = 20sin^2(theta/2).

Therefore, the number of centimeters in the vertical height of the end of the pendulum above its lowest position is equal to 20sin^2(theta/2).

To find the vertical height of the end of the pendulum above its lowest position, we can use trigonometric relationships.

Let's start by visualizing the scenario. The pendulum is swinging from side to side, forming an angle θ with a vertical line (the lowest position). We can define two components of the swinging motion: the horizontal component and the vertical component.

The length of the pendulum is given as 10 cm, which represents the hypotenuse of a right triangle formed by the pendulum's motion. The vertical height we are interested in is the opposite side of this right triangle.

From basic trigonometry, we know that the sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, sin(θ) = opposite / hypotenuse.

sin(θ) = opposite / 10

To find the vertical height in centimeters, we need to isolate the opposite side. Multiplying both sides by 10 gives us:

opposite = 10 * sin(θ)

Now we have the vertical height in terms of the sine function. However, we want to express it in terms of θ.

Recall the double-angle identity for sine: sin(2θ) = 2sin(θ)cos(θ). Rearranging this equation, we can isolate sin(θ):

sin(θ) = (1/2)sin(2θ) / cos(θ)

Since we are interested in sin²(θ/2), we can substitute θ/2 for 2θ:

sin²(θ/2) = (1/2)sin(θ) / cos(θ)

Substituting the previous expression for sin(θ), we get:

sin²(θ/2) = (1/2) * (10 * sin(θ)) / cos(θ)

Now we can replace sin(θ) with the expression given in the problem statement:

sin²(θ/2) = (1/2) * (10 * (20sin²(θ/2))) / cos(θ)

To simplify the equation further, we can multiply all terms by 2 and cancel out common factors:

2sin²(θ/2) = 10 * (20sin²(θ/2)) / cos(θ)

simplifying:

2sin²(θ/2) = 200sin²(θ/2) / cos(θ)

To continue simplifying, we can cancel out sin²(θ/2) on both sides:

2 = 200 / cos(θ)

Finally, we solve for cos(θ):

cos(θ) = 200 / 2
cos(θ) = 100

Therefore, the number of centimeters in the vertical height of the end of the pendulum above its lowest position is given by:

20sin²(θ/2) = 20sin²(θ/2) * cos(θ)
20sin²(θ/2) = 20 * (100/2)
20sin²(θ/2) = 1000

So, the vertical height is 1000 cm above the lowest position of the pendulum.