In `Delta ABC, /_ B = 90^(@), BC = 5cm` and AC - AB = 1 cm. Find the value of sinC and cos C.

Answer

To solve the problem step by step, we will analyze the given information and apply the Pythagorean theorem to find the values of sin C and cos C.

### Step 1: Understand the triangle and given information
We have a right triangle ABC where:
- Angle B = 90°
- Side BC = 5 cm
- AC - AB = 1 cm

### Step 2: Define the sides
Let:
- AB = x cm
- AC = x + 1 cm (since AC - AB = 1 cm)

### Step 3: Apply the Pythagorean theorem
According to the Pythagorean theorem, in a right triangle:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values we have:
\[ (x + 1)^2 = x^2 + 5^2 \]

### Step 4: Expand and simplify the equation
Expanding the left side:
\[ x^2 + 2x + 1 = x^2 + 25 \]
Now, we can simplify this equation by subtracting \( x^2 \) from both sides:
\[ 2x + 1 = 25 \]

### Step 5: Solve for x
Now, isolate x:
\[ 2x = 25 - 1 \]
\[ 2x = 24 \]
\[ x = 12 \]

### Step 6: Find the lengths of AB and AC
Now that we have x:
- AB = x = 12 cm
- AC = x + 1 = 12 + 1 = 13 cm

### Step 7: Calculate sin C
The sine of angle C is given by the ratio of the opposite side (AB) to the hypotenuse (AC):
\[ \sin C = \frac{AB}{AC} = \frac{12}{13} \]

### Step 8: Calculate cos C
The cosine of angle C is given by the ratio of the adjacent side (BC) to the hypotenuse (AC):
\[ \cos C = \frac{BC}{AC} = \frac{5}{13} \]

### Final Answers
- \( \sin C = \frac{12}{13} \)
- \( \cos C = \frac{5}{13} \)

---