In triangle PQR, PQ is 5 units long and PR is 8 units long. If angle 𝑄 is 85∘. Find the size of angle 𝑅 to 3 significant figures.

To solve this problem, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle γ opposite the side of length c, the following relationship holds:

c² = a² + b² - 2ab cos(γ)

In this case, we know the lengths of sides PQ and PR (which we can call a and b), and we know the measure of angle Q (which we can call γ). We want to find the measure of angle R (which we can call α).

First, we can use the Law of Cosines to find the length of side QR (which we can call c).

c² = a² + b² - 2ab cos(γ) c² = 5² + 8² - 258 cos(85) c² = 25 + 64 - 80 cos(85) c² = 89 - 80 cos(85) c = sqrt(89 - 80 cos(85))

Next, we can use the Law of Cosines again to find the measure of angle α.

cos(α) = (b² + c² - a²) / (2bc) cos(α) = (8² + c² - 5²) / (28c) cos(α) = (64 + c² - 25) / (16c) cos(α) = (39 + c²) / (16c)

Finally, we can use the inverse cosine function to find the measure of α in degrees.

α = cos⁻¹[(39 + c²) / (16c)]

This will give you the measure of angle R to three significant figures.