In `DeltaABC, angleB=90^(@),AB=24cmandAC`=26cm` then find the values of the following :
(i) tanA
(ii) cot C
(iii) cosC
(iv) sinC
(v) cosecA
(vi) sec A

Answer

To solve the problem step by step, we will use the properties of right triangles and trigonometric ratios.

### Given:
- Triangle ABC with angle B = 90°
- AB = 24 cm (opposite to angle C)
- AC = 26 cm (hypotenuse)

### Step 1: Find the length of BC using the Pythagorean theorem.
According to the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the known values:
\[ 26^2 = 24^2 + BC^2 \]
Calculating the squares:
\[ 676 = 576 + BC^2 \]
Now, isolate \( BC^2 \):
\[ BC^2 = 676 - 576 \]
\[ BC^2 = 100 \]
Taking the square root:
\[ BC = \sqrt{100} = 10 \, \text{cm} \]

### Step 2: Find \( \tan A \)
Using the definition of tangent:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} \]
For angle A:
- Opposite side = BC = 10 cm
- Adjacent side = AB = 24 cm
Thus,
\[ \tan A = \frac{10}{24} = \frac{5}{12} \]

### Step 3: Find \( \cot C \)
Using the definition of cotangent:
\[ \cot C = \frac{\text{adjacent}}{\text{opposite}} \]
For angle C:
- Adjacent side = AB = 24 cm
- Opposite side = BC = 10 cm
Thus,
\[ \cot C = \frac{24}{10} = \frac{12}{5} \]

### Step 4: Find \( \cos C \)
Using the definition of cosine:
\[ \cos C = \frac{\text{adjacent}}{\text{hypotenuse}} \]
For angle C:
- Adjacent side = AB = 24 cm
- Hypotenuse = AC = 26 cm
Thus,
\[ \cos C = \frac{24}{26} = \frac{12}{13} \]

### Step 5: Find \( \sin C \)
Using the definition of sine:
\[ \sin C = \frac{\text{opposite}}{\text{hypotenuse}} \]
For angle C:
- Opposite side = BC = 10 cm
- Hypotenuse = AC = 26 cm
Thus,
\[ \sin C = \frac{10}{26} = \frac{5}{13} \]

### Step 6: Find \( \csc A \)
Using the definition of cosecant:
\[ \csc A = \frac{1}{\sin A} \]
We know \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \):
- Opposite side = BC = 10 cm
- Hypotenuse = AC = 26 cm
Thus,
\[ \sin A = \frac{10}{26} = \frac{5}{13} \]
So,
\[ \csc A = \frac{1}{\sin A} = \frac{13}{5} \]

### Step 7: Find \( \sec A \)
Using the definition of secant:
\[ \sec A = \frac{1}{\cos A} \]
We know \( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \):
- Adjacent side = AB = 24 cm
- Hypotenuse = AC = 26 cm
Thus,
\[ \cos A = \frac{24}{26} = \frac{12}{13} \]
So,
\[ \sec A = \frac{1}{\cos A} = \frac{13}{12} \]

### Summary of Results:
1. \( \tan A = \frac{5}{12} \)
2. \( \cot C = \frac{12}{5} \)
3. \( \cos C = \frac{12}{13} \)
4. \( \sin C = \frac{5}{13} \)
5. \( \csc A = \frac{13}{5} \)
6. \( \sec A = \frac{13}{12} \)