What are the possible values for the coordinate angles in 3-D space?

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SUMMARY

In 3-D coordinate space, the possible values for the coordinate angles must adhere to specific mathematical constraints. The correct answer to the posed question is option e: the cosines of the angles must be less than (√2/2). This conclusion is derived from the equation (cos(α))^2 + (cos(β))^2 + (cos(γ))^2 = 1, which dictates that the sum of the squared cosines must equal one. Additionally, option c is also considered valid since angles greater than (√2/2) correspond to angles greater than 45°.

PREREQUISITES
  • Understanding of trigonometric functions, specifically cosine.
  • Familiarity with 3-D coordinate geometry.
  • Knowledge of vector representation in mathematics.
  • Ability to manipulate and solve equations involving squares and square roots.
NEXT STEPS
  • Study the properties of cosine functions in trigonometry.
  • Learn about the geometric interpretation of angles in 3-D space.
  • Explore vector mathematics, particularly unit vectors and their applications.
  • Investigate the implications of the cosine rule in various geometric contexts.
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Students studying geometry, mathematicians, and anyone interested in understanding the relationships between angles in three-dimensional space.

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1. Homework Statement [/
In 3-D coordinate space, any two of the coordinate angles must …
Select one:
a. sum to less than 1
b. be greater than 90° but less than 180°
c. each be greater than 45°
d. sum to greater than 90° (if they are both less than 90°).
e. have cosines less than (√2/2).

Homework Equations



(cosα(α))^2 + (cos(β))^2 + (cos(γ))^2 = 1

3. The Attempt at a Solution .

since the sum of the squared cosine of alpha beta and gamma = 1 the answer to me is e reason being if the value of the cosine of the angle is (√2/2) then the square = 0.5 and the sum of two of these angles = 1 therefore the cosine has to be less than (√2/2). c could also be an option since all angles with (√2/2) is greater than 45°. but when i tried with example 150° for both angles the cosine is > than (√2/2) but negative. but when squared it is positive which implies the sum of the two would be greater than 1 and denounces 'c' is 'e' then the required solution.
 
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Consider a simple vector given by cosines (1, 0, 0). This is simply a unit vector directed along the X axis. Is either C or E true for it?
 

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