Question Statement

For optional energy absorption, a solar panel needs to be installed at an angle where it directly faces the sun at noon, which is at a $4 8^{\circ}$ angle from the horizontal. If the panel is placed 2 meters above the ground, calculate the length of the supporting rod required using an inverse sine function.

Background and Explanation

This problem involves right-angled trigonometry, specifically the relationship between an angle and the ratio of the opposite side to the hypotenuse. We use the sine function and its inverse to find a missing side length when the height and angle are known.

Solution

Based on the geometry of the installation:

First, we identify the known and unknown values from the figure:

$A B = Height of panel = 2 m$ (This is the side opposite to the angle)
$B C = Length of supporting rod = x$ (This is the hypotenuse)
$Angle θ = 4 8^{\circ}$

We use the definition of the inverse sine function, which states that the angle is the inverse sine of the ratio of the opposite side to the hypotenuse:

$θ = sin^{- 1} (\frac{A B}{B C})$

Substituting the known values into the equation:

$4 8^{\circ} = sin^{- 1} (\frac{2}{x})$

To solve for $x$ , we take the sine ( $sin$ ) of both sides of the equation:

$sin 4 8^{\circ} = \frac{2}{x}$

Next, we rearrange the equation to isolate $x$ :

$x sin 4 8^{\circ} = 2$

$x = \frac{2}{s i n 4 8 ^{\circ}}$

Using a calculator to find the value of $sin 4 8^{\circ} \approx 0.7431$ :

$x \approx 2.69 m$

The length of the supporting rod required is approximately $2.69 meters$ .

Key Formulas or Methods Used

Sine Ratio: $sin (θ) = \frac{Opposite}{Hypotenuse}$
Inverse Sine Function: $θ = sin^{- 1} (\frac{Opposite}{Hypotenuse})$
Algebraic Rearrangement: Solving for a variable in the denominator.

Summary of Steps

Identify the height ( $2 m$ ) as the opposite side and the rod length ( $x$ ) as the hypotenuse.
Set up the inverse sine equation using the given angle of $4 8^{\circ}$ .
Convert the inverse sine equation into a standard sine equation.
Rearrange the formula to isolate the unknown variable $x$ .
Calculate the final length using the sine value of the angle.

Question Statement

Background and Explanation

Solution

Based on the geometry of the installation:

First, we identify the known and unknown values from the figure:

$A B = Height of panel = 2 m$ (This is the side opposite to the angle)
$B C = Length of supporting rod = x$ (This is the hypotenuse)
$Angle θ = 4 8^{\circ}$

We use the definition of the inverse sine function, which states that the angle is the inverse sine of the ratio of the opposite side to the hypotenuse:

$θ = sin^{- 1} (\frac{A B}{B C})$

Substituting the known values into the equation:

$4 8^{\circ} = sin^{- 1} (\frac{2}{x})$

To solve for $x$ , we take the sine ( $sin$ ) of both sides of the equation:

$sin 4 8^{\circ} = \frac{2}{x}$

Next, we rearrange the equation to isolate $x$ :

$x sin 4 8^{\circ} = 2$

$x = \frac{2}{s i n 4 8 ^{\circ}}$

Using a calculator to find the value of $sin 4 8^{\circ} \approx 0.7431$ :

$x \approx 2.69 m$

The length of the supporting rod required is approximately $2.69 meters$ .

Key Formulas or Methods Used

Sine Ratio: $sin (θ) = \frac{Opposite}{Hypotenuse}$
Inverse Sine Function: $θ = sin^{- 1} (\frac{Opposite}{Hypotenuse})$
Algebraic Rearrangement: Solving for a variable in the denominator.

Summary of Steps

Identify the height ( $2 m$ ) as the opposite side and the rod length ( $x$ ) as the hypotenuse.
Set up the inverse sine equation using the given angle of $4 8^{\circ}$ .
Convert the inverse sine equation into a standard sine equation.
Rearrange the formula to isolate the unknown variable $x$ .
Calculate the final length using the sine value of the angle.

9.3 Q-7

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps

9.3 Q-7

Question Statement

Background and Explanation

Solution

Key Formulas or Methods Used

Summary of Steps