In`DeltaABC,angleB=90^(@)and cosA=(9)/(41)`, then find all other trigonometric ratios for `angleA`.

Answer

To find all the trigonometric ratios for angle A in triangle ABC where angle B is 90 degrees and cos A = 9/41, we can follow these steps:

### Step 1: Identify the sides of the triangle
In triangle ABC:
- Angle B = 90 degrees (right angle)
- Cosine of angle A is given as cos A = adjacent side (base) / hypotenuse.

Here, we can denote:
- Adjacent side (base) = 9
- Hypotenuse = 41

### Step 2: Use the Pythagorean theorem to find the opposite side (perpendicular)
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Perpendicular}^2 + \text{Base}^2 \]

Substituting the known values:
\[ 41^2 = \text{Perpendicular}^2 + 9^2 \]
\[ 1681 = \text{Perpendicular}^2 + 81 \]

Now, solve for the perpendicular:
\[ \text{Perpendicular}^2 = 1681 - 81 \]
\[ \text{Perpendicular}^2 = 1600 \]
\[ \text{Perpendicular} = \sqrt{1600} = 40 \]

### Step 3: Calculate the sine of angle A
The sine of angle A is given by:
\[ \sin A = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
Substituting the values:
\[ \sin A = \frac{40}{41} \]

### Step 4: Calculate the tangent of angle A
The tangent of angle A is given by:
\[ \tan A = \frac{\text{Perpendicular}}{\text{Base}} \]
Substituting the values:
\[ \tan A = \frac{40}{9} \]

### Step 5: Calculate the secant of angle A
The secant of angle A is the reciprocal of the cosine:
\[ \sec A = \frac{1}{\cos A} = \frac{\text{Hypotenuse}}{\text{Base}} \]
Substituting the values:
\[ \sec A = \frac{41}{9} \]

### Step 6: Calculate the cosecant of angle A
The cosecant of angle A is the reciprocal of the sine:
\[ \csc A = \frac{1}{\sin A} = \frac{\text{Hypotenuse}}{\text{Perpendicular}} \]
Substituting the values:
\[ \csc A = \frac{41}{40} \]

### Step 7: Calculate the cotangent of angle A
The cotangent of angle A is the reciprocal of the tangent:
\[ \cot A = \frac{1}{\tan A} = \frac{\text{Base}}{\text{Perpendicular}} \]
Substituting the values:
\[ \cot A = \frac{9}{40} \]

### Summary of Trigonometric Ratios for Angle A
- \( \sin A = \frac{40}{41} \)
- \( \cos A = \frac{9}{41} \)
- \( \tan A = \frac{40}{9} \)
- \( \sec A = \frac{41}{9} \)
- \( \csc A = \frac{41}{40} \)
- \( \cot A = \frac{9}{40} \)