In △PQR,q=420 inches, p=400 inches and ∠P=64∘. Find all possible values of ∠Q, to the nearest degree.

Question

asked Jan 13, 2024 80.2k views

2 Answers

Andrew Davis · Answer 1 · 2024-01-15T19:23:12+0000

Final answer:

To find the possible values of ∠Q, apply the Law of Sines to triangle PQR using the given side lengths and angle. Calculate sinQ and then determine ∠Q, considering both acute and obtuse triangle possibilities.

Step-by-step explanation:

To find the possible values of ∠Q in △PQR, we can use the Law of Sines, which states that the ratio of the length of a side to the sine of its opposite angle is the same for all sides and angles in the triangle. The Law of Sines formula is α/sinA = β/sinB = γ/sinC, where α, β, and γ are the side lengths and A, B, and C are the respective opposite angles.

Given side p = 400 inches, side q = 420 inches, and angle ∠P = 64°, we apply the Law of Sines:

sinQ/q = sinP/p

sinQ/420 = sin(64°)/400

sinQ = (sin(64°)/400) × 420

Calculate the value of sinQ and then use the inverse sine function to find the angle measure of ∠Q. As the triangle could be acute or obtuse, consider both cases.

Trafalgarx · Answer 2 · 2024-01-19T12:53:48+0000

The two possible values of angle Q to the nearest degree are
$70^\circ\) and \(110^\circ$ .

In triangle PQR, we are given:

q = 420 inches (side opposite angle Q)

p = 400 inches (side opposite angle P)

$\angle P = 64^\circ$

We can use the Law of Sines to find angle Q:

$(\sin Q)/(q) = (\sin P)/(p)$

First, find sin P:

$\sin 64^\circ \approx 0.8988$

Now, rearrange the Law of Sines equation to solve for sin Q:

$\sin Q = (q * \sin P)/(p)$

$\sin Q = (420 * 0.8988)/(400)$

$\sin Q \approx 0.9411$

Now, find the possible values of angle Q:

$Q = \sin^(-1)(0.9411)$

$Q \approx 70.02^\circ$

Remember, the Law of Sines gives us two possible angles for Q due to the ambiguity of the sine function. Since angle Q is in a triangle, the sum of angles in a triangle is
$180^\circ$ .