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.

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 side c, the following relationship holds:

c² = a² + b² - 2ab cos(γ)

In triangle PQR, we know that PQ = 5 units, PR = 8 units, and ∠Q = 85°. We want to find ∠R. Let's denote ∠R as γ for simplicity.

We can rearrange the Law of Cosines to solve for cos(γ):

cos(γ) = (a² + b² - c²) / (2ab)

In this case, a = PQ = 5 units, b = PR = 8 units, and c is the side opposite ∠Q, which we don't know. However, we don't need to know c to solve for ∠R.

Substituting the known values into the rearranged Law of Cosines gives:

cos(γ) = (5² + 8² - c²) / (2 * 5 * 8)

We know that cos(85°) = 0.0872 (to four decimal places). So we can set up the equation:

0.0872 = (25 + 64 - c²) / 80

Solving this equation for c² gives:

c² = 25 + 64 - 0.0872 * 80

We can then substitute this value back into the rearranged Law of Cosines to solve for cos(γ):

cos(γ) = (5² + 8² - (25 + 64 - 0.0872 * 80)) / (2 * 5 * 8)

Finally, to find ∠R, we take the inverse cosine (cos⁻¹) of cos(γ).

Please note that the inverse cosine function will give an angle in radians, so you may need to convert to degrees if your calculator is set to give an answer in radians.

This will give you the size of ∠R to three significant figures.