Let's solve the given problem step by step.
### Problem Setup:
We are given:
- [tex]\(\sin A = x + 2\)[/tex]
- [tex]\(\cos B = 2x - 5\)[/tex]
### Step 1: Understanding the Relationship
In a right triangle, the sum of the angles is [tex]\(90^\circ\)[/tex]. Therefore, if [tex]\(\angle A\)[/tex] and [tex]\(\angle B\)[/tex] are acute angles of a right triangle, then:
[tex]\[ \angle A + \angle B = 90^\circ \][/tex]
This implies:
[tex]\[ \sin A = \cos B \][/tex]
### Step 2: Equating the Given Expressions
From the given information:
[tex]\[ \sin A = x + 2 \][/tex]
[tex]\[ \cos B = 2x - 5 \][/tex]
Since [tex]\( \sin A = \cos B \)[/tex]:
[tex]\[ x + 2 = 2x - 5 \][/tex]
### Step 3: Solving for [tex]\( x \)[/tex]
Let's isolate [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ x + 2 = 2x - 5 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 2 = x - 5 \][/tex]
Now, add 5 to both sides:
[tex]\[ 2 + 5 = x \][/tex]
Combine the constants:
[tex]\[ 7 = x \][/tex]
### Step 4: Checking the Validity
Given the options are:
- [tex]\( x = 2.33 \)[/tex]
- [tex]\( x = 29 \)[/tex]
- [tex]\( x = 7 \)[/tex]
- [tex]\( x = -7 \)[/tex]
The value we've found, [tex]\( x = 7 \)[/tex], is among the options.
### Conclusion
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ \boxed{7} \][/tex]
