`DeltaDEF` is right angled at E. If `mangleF=30^(@)` , then find the value of `(sinD-(1)/(3))`.

Options

A. `(-1)/(2sqrt(3))`
B. `((3sqrt(3)-2))/(6)`
C. `((sqrt(2)-sqrt(3)))/(sqrt(6))`
D. `((2sqrt(2)-1))/(sqrt(2))`

Answer

To solve the problem step by step, we will analyze the right triangle DEF, where angle E is the right angle and angle F is given as 30 degrees.

### Step 1: Identify the angles in triangle DEF
Since triangle DEF is a right triangle with angle E = 90°, and angle F = 30°, we can find angle D using the fact that the sum of angles in a triangle is 180°.

\[ \text{Angle D} = 180° - \text{Angle E} - \text{Angle F} = 180° - 90° - 30° = 60° \]

### Step 2: Find sin D
Now that we know angle D is 60°, we can find sin D.

\[ \sin D = \sin 60° = \frac{\sqrt{3}}{2} \]

### Step 3: Calculate sin D - (1/3)
Next, we need to calculate the expression \( \sin D - \frac{1}{3} \).

\[ \sin D - \frac{1}{3} = \frac{\sqrt{3}}{2} - \frac{1}{3} \]

### Step 4: Find a common denominator
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6.

\[ \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{6} \quad \text{(multiply numerator and denominator by 3)} \]
\[ \frac{1}{3} = \frac{2}{6} \quad \text{(multiply numerator and denominator by 2)} \]

### Step 5: Perform the subtraction
Now we can substitute back into the expression:

\[ \sin D - \frac{1}{3} = \frac{3\sqrt{3}}{6} - \frac{2}{6} = \frac{3\sqrt{3} - 2}{6} \]

### Final Answer
Thus, the value of \( \sin D - \frac{1}{3} \) is:

\[ \frac{3\sqrt{3} - 2}{6} \]

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