Determine the expression that represents `mangleR` in triangle PQR given that side PQ=7, side QR=9, and `mangleP`=110

Options

A. `sin^(1)((7sin110^(@))/(9))`
B. `sin^(-1)63sin110^(@)`
C. `sin^(-1)((sin)(63)^(@))`
D. `(7sin110^(@))/(63)`

Answer

To determine the expression that represents angle R in triangle PQR, we will use the information given: side PQ = 7, side QR = 9, and angle P = 110 degrees. We will apply the sine rule and some trigonometric identities to find angle R.

### Step-by-Step Solution:

1. **Draw the Triangle**:
First, we draw triangle PQR with points P, Q, and R. Label the sides as follows:
- PQ = 7
- QR = 9
- Angle P = 110 degrees

2. **Drop a Perpendicular**:
Drop a perpendicular from point Q to side PR, and label the foot of the perpendicular as point O. This creates two right triangles: triangle OPQ and triangle OQR.

3. **Identify the Right Triangle**:
In triangle OPQ, we can identify:
- OP = PQ * cos(P) = 7 * cos(110°)
- OQ = PQ * sin(P) = 7 * sin(110°)

4. **Calculate OQ**:
We can calculate OQ using the sine function:
\[ OQ = 7 \cdot \sin(110°) \]

5. **Use the Sine Rule in Triangle OQR**:
In triangle OQR, we have:
- OQ is the opposite side to angle R.
- QR is the hypotenuse.

By the definition of sine:
\[ \sin R = \frac{OQ}{QR} \]
Substituting the values we have:
\[ \sin R = \frac{7 \cdot \sin(110°)}{9} \]

6. **Find Angle R**:
To find angle R, we take the inverse sine:
\[ R = \sin^{-1}\left(\frac{7 \cdot \sin(110°)}{9}\right) \]

### Final Expression:
Thus, the expression that represents angle R in triangle PQR is:
\[ R = \sin^{-1}\left(\frac{7 \cdot \sin(110°)}{9}\right) \]