Find the third directional cosine (there are two of them with opposite signs) if it is known that two directional cosines are equal to 1/2 and 1/3

Answer:

To find the third directional cosine, we need to use the property that the sum of the squares of the directional cosines is equal to 1.

Let's denote the three directional cosines as cosα, cosβ, and cosγ. Given that two directional cosines are equal to 1/2 and 1/3, we can set up the following equations:

cosα = 1/2

cosβ = 1/3

To find cosγ, we can rearrange the equation for the sum of the squares of the directional cosines:

cos²α + cos²β + cos²γ = 1

Substituting the given values, we have:

(1/2)² + (1/3)² + cos²γ = 1

Simplifying the equation, we get:

1/4 + 1/9 + cos²γ = 1

To solve for cos²γ, we can combine the fractions:

(9/36) + (4/36) + cos²γ = 1

(13/36) + cos²γ = 1

Now, we can solve for cos²γ by subtracting 13/36 from both sides:

cos²γ = 1 - 13/36

cos²γ = 23/36

Taking the square root of both sides, we find:

cosγ = √(23/36)

Since we are looking for the third directional cosine, there are two possible solutions, one positive and one negative. So, the third directional cosine can be either √(23/36) or -√(23/36).

In conclusion, the third directional cosine, cosγ, is either √(23/36) or -√(23/36).