In a triangle PQR, angle Q=50 degrees,PR=5CM, QR=4CM. find angle P?
Looks like the "ambiguous case" , or sometimes called the SSA setup, where A is NOT the contained angle
sinP/4 = sin 50°/5
sin P = 4sin50/5 = .61284
angle P = 37.8° or 180-37.8 ° which is 142.2°
but the second case would result in (angleR + 142.2+50) > 180, thus that case is not admissable.
so angle P is 37.8°
To find angle P in triangle PQR, we can use the fact that the sum of the angles of a triangle is always 180 degrees.
Step 1: Find the measure of angle R.
Since we know that angle Q = 50 degrees, angle R + angle Q + angle P = 180 degrees.
Substituting the given values, we have:
angle R + 50 degrees + angle P = 180 degrees.
Step 2: Solve for angle R.
To isolate angle R, we need to rearrange the equation:
angle R = 180 degrees - 50 degrees - angle P
Step 3: Substitute the given side lengths into the Law of Cosines formula.
Since we are given the side lengths PR = 5cm and QR = 4cm, we can use the Law of Cosines formula to find the measure of angle P.
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a = PR = 5cm, b = QR = 4cm, and c is the side opposite angle P (unknown).
Substituting the given values, we have:
c^2 = 5^2 + 4^2 - 2 * 5 * 4 * cos(P)
Step 4: Solve for angle P.
Using the Law of Cosines, we can solve for cos(P) and then find angle P.
c^2 = 5^2 + 4^2 - 2 * 5 * 4 * cos(P)
Note that c^2 is the unknown side length opposite angle P.
Step 5: Substitute the value of c into the equation from Step 2.
Now, substitute the value of c into the equation from Step 2 in order to solve for angle R:
angle R = 180 degrees - 50 degrees - angle P
By following these steps, you can find the value of angle P in triangle PQR.