How will you find the distance of a star from Sun, given that the angle at which it is viewed from the two solstices of Earth(i.e winter and summer) are alpha and beta respectively, and the distance between the points of the solstices is 150 million km.
{Hint - form a triangle using the given data. draw a perpendicular, find the height of the perpendicular
To find the distance of a star from the Sun using the given information, we can follow these steps:
Step 1: Draw a diagram
Start by drawing a diagram to visualize the information provided. Draw a straight line to represent the distance between the points of the solstices (150 million km). Label one end of the line as the Winter solstice and the other end as the Summer solstice. Next, draw a star above the line to represent the star whose distance we want to find. Finally, draw two lines from the star to each solstice, forming a triangle.
Step 2: Label the angles
Label the angle between the line from the star to the Winter solstice and the line from the star to the Summer solstice as "α." Similarly, label the angle between the line from the star to the Summer solstice and the line from the star to the Winter solstice as "β."
Step 3: Label the perpendicular distance
Draw a perpendicular line from the star down to the line representing the distance between the solstices. Label this perpendicular distance as "h" (height).
Step 4: Use trigonometry to find the height
Since we know the angle α and the length of the side opposite to it (h), we can use trigonometry to determine the value of h. We will use the tangent function, which relates the opposite and adjacent sides of a right triangle.
The formula for tangent is: tan(α) = opposite/adjacent.
In this case, h is the opposite side and the distance between the solstices is the adjacent side. Substitute the known values into the formula and solve for h.
Step 5: Use the Pythagorean theorem to find the distance to the star
Now that we have the height (h) of the perpendicular, we can use the Pythagorean theorem to find the distance between the star and the Sun. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the length of the hypotenuse is the distance between the star and the Sun. The two other sides are the perpendicular height (h) and the distance between the solstices (150 million km). Use the formula:
Distance to the star^2 = h^2 + (150 million km)^2
Take the square root of the result to find the distance to the star.
By following these steps and using trigonometry and the Pythagorean theorem, we can determine the distance of the star from the Sun based on the given information.