`Delta PQR` is right angled at Q. If ` angle R = 30^(@),` then find the value of (cos P -1/3).

Options

A. `1//6`
B. `(2 sqrt2 -1) sqrt2`
C. `-1 //sqrt3`
D. `(sqrt3 -4 ) //2 sqrt3`

Answer

To solve the problem step by step, we will analyze the triangle PQR, which is right-angled at Q and has angle R equal to 30 degrees. We need to find the value of (cos P - 1/3).

### Step 1: Identify the angles in triangle PQR
Since triangle PQR is right-angled at Q, we know:
- Angle Q = 90 degrees
- Angle R = 30 degrees

Using the property that the sum of angles in a triangle equals 180 degrees, we can find angle P.

### Step 2: Calculate angle P
Using the angle sum property:
\[ \text{Angle P} + \text{Angle Q} + \text{Angle R} = 180^\circ \]
Substituting the known values:
\[ \text{Angle P} + 90^\circ + 30^\circ = 180^\circ \]
\[ \text{Angle P} + 120^\circ = 180^\circ \]
\[ \text{Angle P} = 180^\circ - 120^\circ = 60^\circ \]

### Step 3: Calculate cos P
Now that we have angle P, we can find cos P:
\[ \text{cos P} = \text{cos}(60^\circ) \]
From trigonometric values, we know:
\[ \text{cos}(60^\circ) = \frac{1}{2} \]

### Step 4: Substitute into the expression (cos P - 1/3)
Now we substitute cos P into the expression:
\[ \text{cos P} - \frac{1}{3} = \frac{1}{2} - \frac{1}{3} \]

### Step 5: Find a common denominator and simplify
To subtract these fractions, we need a common denominator. The least common multiple of 2 and 3 is 6.
\[ \frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{1}{3} = \frac{2}{6} \]
Now we can perform the subtraction:
\[ \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

### Final Answer
Thus, the value of (cos P - 1/3) is:
\[ \frac{1}{6} \]

---