Answer:
- ∠P = 19°
- ∠R = 71°
- RP = 9.41
Step-by-step explanation:
You want the unknown angles and hypotenuse in right triangle PQR with Q=90°, PQ=8.9, and QR=3.1.
Trig relations
The relations between trig functions and side lengths in a right triangle are represented by the mnemonic SOH CAH TOA. For the purpose here, we will make use of two of these relations:
Tan = Opposite/Adjacent ⇒ tan(P) = QR/PQ
Cos = Adjacent/Hypotenuse ⇒ cos(P) = PQ/RP
Angles
Using the tangent relation, we have ...
tan(P) = 3.1/8.9
P = arctan(3.1/8.9) ≈ 19°
Angle R is the complement of angle P, so is ...
R = 90° -P = 90° -19° = 71°
The two angles are ...
Hypotenuse
Using the cosine relation, we can find the length of RP.
cos(P) = PQ/RP
RP = PQ/cos(P) = 8.9/cos(19.204°) ≈ 9.42
The measure of RP is about 9.42 units.
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Additional comment
Ordinarily, you might use the Pythagorean theorem to find RP:
RP = √(QR² +PQ²) = √(3.1² +8.9²)
RP = √(9.61+79.21) = √88.82 ≈ 9.42
We believe the method used above takes less effort with the calculator.
