Math / Trigonometry

QuestionQuestion 5
Consider a triangle $A B C$ like the one below. Suppose that $A=127^{\circ}, b=37$ , and $c=22$ . (The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.
If there is more than one solution, use the button labeled "or".

Studdy Solution

STEP 1

1. We are given a triangle $\triangle ABC$ with angle $A = 127^\circ$ , side $b = 37$ , and side $c = 22$ .
2. We need to find the remaining sides and angles of the triangle.
3. The Law of Sines and the Law of Cosines will be used for calculations.
4. Solutions should be rounded to the nearest tenth.

STEP 2

1. Use the Law of Cosines to find side $a$ .
2. Use the Law of Sines to find angle $B$ .
3. Calculate angle $C$ .
4. Verify if there is more than one solution.

STEP 3

Use the Law of Cosines to find side $a$ :
$a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$
Substitute the given values:
$a^2 = 37^2 + 22^2 - 2 \times 37 \times 22 \times \cos(127^\circ)$
Calculate $\cos(127^\circ)$ and then solve for $a$ :
$a^2 = 1369 + 484 + 2 \times 37 \times 22 \times \cos(127^\circ)$
$a^2 = 1853 + 1628 \times (-0.6018)$
$a^2 = 1853 - 979.3$
$a^2 = 2872.3$
$a = \sqrt{2872.3}$
$a \approx 53.6$

STEP 4

Use the Law of Sines to find angle $B$ :
$\frac{\sin(B)}{b} = \frac{\sin(A)}{a}$
$\sin(B) = \frac{b \cdot \sin(A)}{a}$
Substitute the known values:
$\sin(B) = \frac{37 \cdot \sin(127^\circ)}{53.6}$
Calculate $\sin(127^\circ)$ and solve for $B$ :
$\sin(B) = \frac{37 \cdot 0.7986}{53.6}$
$\sin(B) = \frac{29.5542}{53.6}$
$\sin(B) \approx 0.5513$
Find $B$ using the inverse sine function:
$B \approx \sin^{-1}(0.5513)$
$B \approx 33.5^\circ$

STEP 5

Calculate angle $C$ using the angle sum property of triangles:
$C = 180^\circ - A - B$
$C = 180^\circ - 127^\circ - 33.5^\circ$
$C \approx 19.5^\circ$

STEP 6

Verify if there is more than one solution:
Since angle $A$ is obtuse, there is only one solution for the triangle.
The solved triangle has the following dimensions:
$a \approx 53.6$ $B \approx 33.5^\circ$ $C \approx 19.5^\circ$

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