In a right triangle, `sinx^(@)=cosy^(@)`. If `x=3c+14 and y=7c+11`, what is the value of c?

Options

A. `3.5`
B. `6.5`
C. `11.5`
D. `22.5`

Answer

To solve the problem, we start with the given equation and relationships.

### Step 1: Use the property of trigonometric functions
We know that in a right triangle, \( \sin x = \cos y \) can be rewritten using the complementary angle property:
\[ \sin x = \cos y \implies x + y = 90^\circ \]

### Step 2: Substitute the expressions for \(x\) and \(y\)
Given \(x = 3c + 14\) and \(y = 7c + 11\), we can substitute these into the equation:
\[ (3c + 14) + (7c + 11) = 90 \]

### Step 3: Simplify the equation
Combine like terms:
\[ 3c + 7c + 14 + 11 = 90 \]
\[ 10c + 25 = 90 \]

### Step 4: Solve for \(c\)
Now, isolate \(c\) by first subtracting 25 from both sides:
\[ 10c = 90 - 25 \]
\[ 10c = 65 \]
Now, divide both sides by 10:
\[ c = \frac{65}{10} = 6.5 \]

### Final Answer
Thus, the value of \(c\) is:
\[ c = 6.5 \]