`DeltaXYZ` is right angled at Y . If `mangleX=60^(@)` , then find the value of `(SecZ+(2)/(sqrt(3)))`

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the angles in the triangle
Given that triangle XYZ is right-angled at Y, we know that:
- Angle Y = 90°
- Angle X = 60°

Using the property that the sum of the angles in a triangle is 180°, we can find angle Z.

### Step 2: Calculate angle Z
Using the formula:
\[ \text{Angle X} + \text{Angle Y} + \text{Angle Z} = 180° \]
Substituting the known values:
\[ 60° + 90° + \text{Angle Z} = 180° \]
\[ \text{Angle Z} = 180° - 150° = 30° \]

### Step 3: Find the value of sec Z
The secant function is defined as:
\[ \sec Z = \frac{1}{\cos Z} \]
For angle Z = 30°, we know that:
\[ \cos 30° = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sec 30° = \frac{1}{\cos 30°} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 4: Calculate the expression Sec Z + \frac{2}{\sqrt{3}}
Now we substitute the value of sec Z into the expression:
\[ \sec Z + \frac{2}{\sqrt{3}} = \frac{2}{\sqrt{3}} + \frac{2}{\sqrt{3}} = \frac{2 + 2}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \sec Z + \frac{2}{\sqrt{3}} \) is:
\[ \frac{4}{\sqrt{3}} \]

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