Distance between the points (a cos 35°,0) and (0,a cos 65°)

To find the distance between the two given points, we can use the distance formula. The distance formula calculates the straight-line distance between two points in a coordinate plane.

The Distance Formula:
The distance formula is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

The distance formula between two points (x1, y1) and (x2, y2) is given by:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, the given points are (a cos 35°, 0) and (0, a cos 65°). Let's calculate the distance between them.

Calculating the Distance:
We will substitute the coordinates of the given points into the distance formula and solve for the distance.

Point 1 = (a cos 35°, 0)
Point 2 = (0, a cos 65°)

Using the distance formula, we have:

d = sqrt((0 - a cos 35°)^2 + (a cos 65° - 0)^2)

Simplifying further:

d = sqrt((a cos 35°)^2 + (a cos 65°)^2)

Applying Trigonometric Identities:
We can simplify the equation by using trigonometric identities. The identity cos^2θ + sin^2θ = 1 can be applied here.

Since cos^2θ = 1 - sin^2θ, we can rewrite the equation as:

d = sqrt(a^2 - a^2 sin^2 35° + a^2 - a^2 sin^2 65°)

Simplifying further:

d = sqrt(a^2(1 - sin^2 35°) + a^2(1 - sin^2 65°))

d = sqrt(a^2 - a^2 sin^2 35° + a^2 - a^2 sin^2 65°)

d = sqrt(2a^2 - a^2 sin^2 35° - a^2 sin^2 65°)

This is the simplified form of the distance between the given points (a cos 35°,0) and (0,a cos 65°). The distance is represented by the variable 'd' and is equal to sqrt(2a^2 - a^2 sin^2 35° - a^2 sin^2 65°).