In triangle PQR, |PQ|= 7 cm, |QR|=5 cm and |<QPR|= 43 degrees Find, correct to one decimal place, the value, or values, of |<QPR|
you give |<QPR|= 43°, then ask for |<QPR|.
assuming you want to know the rest of the values for the triangle, use the law of sines:
sin43°/5 = sinR/7
R = 72.7°
so, angle Q = 180-(43+72.7) = 64.3°
all we have left is side q, so back to the law of sines:
q/sin64.3° = 5/sin43°
q = 6.6
To find the value of angle ∠QPR, we can use the Law of Cosines.
The Law of Cosines states that for any triangle with sides a, b, and c, and the angle between sides a and b denoted as ∠C, the following equation holds:
c^2 = a^2 + b^2 - 2abcos(C)
In our case, we know that |PQ| = 7 cm, |QR| = 5 cm, and we want to find ∠QPR.
Let's substitute the given values into the Law of Cosines formula:
|QR|^2 = |PQ|^2 + |QR|^2 - 2|PQ||QR|cos(∠QPR)
5^2 = 7^2 + 5^2 - 2(7)(5)cos(∠QPR)
25 = 49 + 25 - 70cos(∠QPR)
25 = 74 - 70cos(∠QPR)
Simplifying further:
70cos(∠QPR) = 74 - 25
70cos(∠QPR) = 49
cos(∠QPR) = 49/70
Now, to find the value of ∠QPR, we can use the inverse cosine function (cos^(-1)):
∠QPR = cos^(-1)(49/70)
Using a calculator, we can find the value of ∠QPR as:
∠QPR ≈ 32.1 degrees
Therefore, the value of angle ∠QPR is approximately 32.1 degrees.