What is the missing reason in the proof? corresponding angles theorem alternate interior angles theorem vertical angles theorem alternate exterior angles theorem
V2 csc*x+cscx=v2
The Correct Answer and Explanation is :
1. Missing Reason in the Proof:
To determine the missing reason, let’s evaluate the potential scenarios. Theorems listed correspond to specific relationships in geometry proofs:
- Corresponding Angles Theorem: States that corresponding angles are congruent when a transversal crosses two parallel lines.
- Alternate Interior Angles Theorem: States that alternate interior angles are congruent when a transversal crosses two parallel lines.
- Vertical Angles Theorem: States that vertical angles (formed by two intersecting lines) are congruent.
- Alternate Exterior Angles Theorem: States that alternate exterior angles are congruent when a transversal crosses two parallel lines.
If the proof involves parallel lines and a transversal, it’s likely Corresponding Angles Theorem or Alternate Interior/Exterior Angles Theorem is the missing reason. If two intersecting lines are involved, Vertical Angles Theorem is likely the missing reason. If you provide the exact setup, I can identify the specific theorem.
2. Solve ( \sqrt{2} \csc(x) + \csc(x) = \sqrt{2} ):
Solution:
- Combine like terms:
( (\sqrt{2} + 1) \csc(x) = \sqrt{2} ). - Solve for ( \csc(x) ):
( \csc(x) = \frac{\sqrt{2}}{\sqrt{2} + 1} ). - Rationalize the denominator:
Multiply numerator and denominator by ( \sqrt{2} – 1 ):
( \csc(x) = \frac{\sqrt{2}(\sqrt{2} – 1)}{(\sqrt{2} + 1)(\sqrt{2} – 1)} ).
Simplify:
( \csc(x) = \frac{2 – \sqrt{2}}{1} = 2 – \sqrt{2} ). - Rewrite as ( \csc(x) = 2 – \sqrt{2} ).
Explanation (300 words):
The given equation ( \sqrt{2} \csc(x) + \csc(x) = \sqrt{2} ) involves trigonometric functions. Simplifying requires combining coefficients of ( \csc(x) ). Recognizing ( \csc(x) = \frac{1}{\sin(x)} ), solving for ( \csc(x) ) helps determine sine’s value.
Rationalization simplifies fractions by removing radicals from denominators. Multiplying by the conjugate ensures calculations are precise and fractions are in standard form. Solving gives ( \csc(x) = 2 – \sqrt{2} ). This result confirms the reciprocal relation: ( \sin(x) = \frac{1}{\csc(x)} ).
Such proofs apply widely in trigonometry, demonstrating equation-solving techniques and properties of trig functions, useful for understanding waves, oscillations, and geometry.