Answer:
- ∠P = 27.4°
- ∠Q = 57.4°
- ∠R = 95.2°
- CD = 12.4
- ED = 5.7
- ∠D = 42°
Explanation:
You want the solutions to the triangles shown using the law of cosines and the law of sines.
Law of Cosines
The law of cosines tells you the relation between two sides and the angle between them:
c² = a² +b² -2ab·cos(C)
Solving for the angle gives ...
C = arccos((a² +b² -c²)/(2ab))
Law of Sines
The law of sines tells you side lengths are proportional to the sines of their opposite angles.
a/sin(A) = b/sin(B) = c/sin(C)
Knowing an opposite pair of side and angle, we can solve for the other sides or angles by rearranging:
b = (a/sin(A))·sin(B)
B = arcsin(b·sin(A)/a)
Sum of angles
The sum of angles in a triangle is 180°, so we can always find the third angle once we know two of them.
a. Sides given
We like to start by finding the largest angle, the one opposite the longest side. Here, that is ...
R = arccos((6² +11² -13²)/(2·6·11)) ≈ 95.2°
The law of sines tells us another ange:
Q = arcsin(11·sin(95.2°)/13) ≈ 57.4°
P = 180° -95.2° -57.4° = 27.4°
The solution is (P, Q, R) = (27.4°, 57.4°, 95.2°).
b. Angles given
Before we can use the law of sines, we need a side-angle pair. The only given side is opposite a missing angle, so we need to find that angle first.
D = 180° -113° -25° = 42°
Then the other sides can be found.
CD = sin(113°)·9/sin(42°) ≈ 12.4
ED = sin(25°)·9/sin(42°) ≈ 5.7
The solution is (CD, ED, D) = (12.4, 5.7, 42°).
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Additional comment
Solving the law of cosines formula for the missing side gives ...
c = √(a² +b² -2ab·cos(C))
As you can see in the first attachment, these formulas are easily evaluated in one step using a suitable calculator. Intermediate values should always be preserved at full precision. Rounding should only be done on the final answers.
The last two attachments show an online triangle solver's solution to these problems. Some calculators have a triangle solver app built in. Stand-alone solver apps are also available.
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