Question

In △ RST, r=4.3 inches, s=8.9 inches and t=6.5 inches. Find the measure of ∠ R to the nearest 10th of a degree.

Answer 1

The measure of angle R in triangle RST is approximately
`\( 27.0^\circ \)` (to the nearest tenth of a degree) using the Law of Cosines.

To find the measure of angle R in triangle RST , we can use the Law of Cosines, which states that for any triangle with sides a, b, and c and an angle C opposite side c :

`\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]`

In this case, r = 4.3 inches, s = 8.9 inches, and t = 6.5 inches. Let's denote R as the angle opposite side r , so c = r . We are solving for angle R :

`\[ 4.3^2 = 8.9^2 + 6.5^2 - 2 \cdot 8.9 \cdot 6.5 \cos(R) \]`

Now, rearrange and solve for cos(R) :

`\[ \cos(R) = (8.9^2 + 6.5^2 - 4.3^2)/(2 \cdot 8.9 \cdot 6.5) \]`

Once you find
`\( \cos(R) \)`), take the arccosine to find R in radians. Finally, convert radians to degrees. The result is the measure of angle R .

We find that
`\( \cos(R) = (8.9^2 + 6.5^2 - 4.3^2)/(2 \cdot 8.9 \cdot 6.5) \).`

`\[ \cos(R) = (79.21 + 42.25 - 18.49)/(115.7) \]`

`\[ \cos(R) = (102.97)/(115.7) \]`

`\[ \cos(R) \approx 0.8897 \]`

Now,
`\( R = \cos^(-1)(0.8897) \) in radians. Using a calculator, \( R \approx 0.4717 \) radians.`

To convert radians to degrees, multiply by
`\( (180)/(\pi) \):`

`\[ R \approx 0.4717 * (180)/(\pi) \]`

`\[ R \approx 27.03^\circ \]`

Therefore, the measure of angle R is approximately
`\( 27.0^\circ \)` (to the nearest tenth of a degree).