`DeltaDEF` is right angled at E . If `mangleD=30^(@)` , find the value of `(cosF+sqrt(3))` .

Options

A. `(7)/(2sqrt(3))`
B. `((1+2sqrt(3)))/(2)`
C. `(4)/(sqrt(3))`
D. `((sqrt(3)+4))/(2)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the triangle
We have a triangle DEF which is right-angled at point E. We know that angle D is 30 degrees.

### Step 2: Calculate angle F
In any triangle, the sum of all angles is 180 degrees. Therefore, we can express this relationship as:
\[ \text{Angle D} + \text{Angle E} + \text{Angle F} = 180^\circ \]

Substituting the known values:
\[ 30^\circ + 90^\circ + \text{Angle F} = 180^\circ \]

### Step 3: Solve for angle F
Now, we can simplify the equation:
\[ 120^\circ + \text{Angle F} = 180^\circ \]
Subtracting 120 degrees from both sides gives us:
\[ \text{Angle F} = 180^\circ - 120^\circ = 60^\circ \]

### Step 4: Calculate cos F
Now that we know angle F is 60 degrees, we can find the cosine of angle F:
\[ \cos F = \cos 60^\circ \]

From trigonometric values, we know:
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 5: Add root 3 to cos F
Now we need to find the value of \( \cos F + \sqrt{3} \):
\[ \cos F + \sqrt{3} = \frac{1}{2} + \sqrt{3} \]

### Final Answer
Thus, the value of \( \cos F + \sqrt{3} \) is:
\[ \frac{1}{2} + \sqrt{3} \]