In`DeltaABC,angleA=0=90^(@),AB=6cmandAC=8cm`, then find the values of the following :
(i) sinB
(ii)sinA
(iii) tanB
(iv)secB
(v)cotC
(vi) "cosec" C

Answer

To solve the problem step by step, we will analyze triangle ABC where angle A = 90°, AB = 6 cm, and AC = 8 cm. We need to find the values of sin B, sin A, tan B, sec B, cot C, and cosec C.

### Step 1: Draw Triangle ABC
- Draw triangle ABC with angle A as 90°.
- Label the sides: AB = 6 cm (adjacent to angle B), AC = 8 cm (opposite to angle B), and BC as the hypotenuse.

### Step 2: Find the Length of Hypotenuse BC
Using the Pythagorean theorem:
\[ BC^2 = AB^2 + AC^2 \]
Substituting the values:
\[ BC^2 = 6^2 + 8^2 = 36 + 64 = 100 \]
Taking the square root:
\[ BC = \sqrt{100} = 10 \text{ cm} \]

### Step 3: Calculate sin B
Using the definition of sine:
\[ \sin B = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AC}{BC} = \frac{8}{10} = \frac{4}{5} \]

### Step 4: Calculate sin A
Since angle A is 90°:
\[ \sin A = \sin 90° = 1 \]

### Step 5: Calculate tan B
Using the definition of tangent:
\[ \tan B = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AC}{AB} = \frac{8}{6} = \frac{4}{3} \]

### Step 6: Calculate sec B
Using the definition of secant:
\[ \sec B = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{BC}{AB} = \frac{10}{6} = \frac{5}{3} \]

### Step 7: Calculate cot C
Using the definition of cotangent:
\[ \cot C = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{AB}{AC} = \frac{6}{8} = \frac{3}{4} \]

### Step 8: Calculate cosec C
Using the definition of cosecant:
\[ \csc C = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{BC}{AC} = \frac{10}{8} = \frac{5}{4} \]

### Summary of Results
- (i) sin B = \(\frac{4}{5}\)
- (ii) sin A = \(1\)
- (iii) tan B = \(\frac{4}{3}\)
- (iv) sec B = \(\frac{5}{3}\)
- (v) cot C = \(\frac{3}{4}\)
- (vi) cosec C = \(\frac{5}{4}\)