Answer:
∠T = 42°
PT = 14.4
QT = 19.4
Explanation:
An angle and its adjacent side are given in the right triangle. The other two sides can be found using trig relations. The third angle can be found using the angle sum theorem.
Trig relations
The mnemonic SOH CAH TOA is intended to remind you of the relations between sides of a right triangle and the trig functions of its angles. The relations relevant to this triangle are ...
Cos = Adjacent/Hypotenuse ⇒ Hypotenuse = Adjacent/Cos
Tan = Opposite/Adjacent ⇒ Opposite = Adjacent×Tan
Then the missing side lengths are ...
QT = QP/cos(Q) = 13/cos(48°) ≈ 19.4282 ≈ 19.4
PT = PQ×tan(Q) = 13×tan(48°) ≈ 14.43796 ≈ 14.4
Angle sum
The angle sum theorem tells you the sum of angles in a triangle is 180°. This means the acute angles in a right triangle are complementary.
∠T = 90° -48° = 42°
The solution to the triangle is ...
∠T = 42°
PT = 14.4
QT = 19.4
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Additional comment
The attachment shows the solution from one of many available triangle solvers. Some calculators have triangle solver functions built in.