QuestionSolve the triangle. (Do not round until the final answer. Then round to the nearest tenth as needed.) (Do not round until the final answer. Then round to the nearest degree as needed.) A = (Do not round until the final answer. Then round to the nearest degree as needed.)
Studdy Solution
STEP 1
1. We are given a right triangle (B = 90°).
2. Side a = 8 units.
3. Side c = 2 units.
4. We need to find side b, angle C, and angle A.
5. We should not round intermediate calculations.
6. Final answers should be rounded to the nearest tenth for lengths and nearest degree for angles.
STEP 2
1. Find side b using the Pythagorean theorem.
2. Calculate angle C using trigonometry.
3. Calculate angle A using the sum of angles in a triangle.
STEP 3
To find side b, we can use the Pythagorean theorem: a² + c² = b²
Substituting the known values:
8² + 2² = b²
64 + 4 = b²
68 = b²
To solve for b, we take the square root of both sides:
b = √68
STEP 4
To find angle C, we can use the inverse tangent function (arctan or tan⁻¹).
In a right triangle, tan(C) = opposite / adjacent = c / a
tan(C) = 2 / 8 = 1/4
C = tan⁻¹(1/4)
Using a calculator (without rounding):
C ≈ 14.0362434679265
STEP 5
To find angle A, we use the fact that the sum of angles in a triangle is 180°:
A + B + C = 180°
A + 90° + 14.0362434679265° = 180°
A = 180° - 90° - 14.0362434679265°
A = 75.9637565320735°
Now, let's provide the final answers with appropriate rounding:
b = √68 ≈ 8.2 (rounded to the nearest tenth)
C ≈ 14° (rounded to the nearest degree)
A ≈ 76° (rounded to the nearest degree)
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