`triangleDEF` is right angled at E. If cotD = 5/12, then what is the value of sinF ?

Options

A. `5//12`
B. `13//5`
C. `5//13`
D. `13//12`

Answer

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

### Step 1: Understand the triangle configuration
We have a triangle DEF which is right-angled at E. This means that angle E is 90 degrees. We need to find the value of sin F given that cot D = 5/12.

### Step 2: Recall the definition of cotangent
Cotangent (cot) is defined as the ratio of the adjacent side to the opposite side in a right triangle. Therefore, if cot D = 5/12, we can interpret this as:
- Adjacent side (to angle D) = 12
- Opposite side (to angle D) = 5

### Step 3: Identify the sides of the triangle
In triangle DEF:
- DE (adjacent to angle D) = 12
- EF (opposite to angle D) = 5

### Step 4: Use the Pythagorean theorem to find the hypotenuse
According to the Pythagorean theorem:
\[ DF^2 = DE^2 + EF^2 \]
Substituting the values we have:
\[ DF^2 = 12^2 + 5^2 \]
\[ DF^2 = 144 + 25 \]
\[ DF^2 = 169 \]
Taking the square root:
\[ DF = \sqrt{169} = 13 \]

### Step 5: Find sin F
The sine of angle F is defined as the ratio of the length of the side opposite angle F to the length of the hypotenuse:
\[ \sin F = \frac{\text{Opposite to F}}{\text{Hypotenuse}} \]
In triangle DEF, the side opposite to angle F is DE (which is 12) and the hypotenuse is DF (which is 13):
\[ \sin F = \frac{EF}{DF} = \frac{12}{13} \]

### Final Answer
Thus, the value of sin F is:
\[ \sin F = \frac{12}{13} \]

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