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Chapter 6: Problem 16

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$B=80^{\circ}, C=10^{\circ}, a=8$$

Short Answer

Expert verified
The solutions for the triangle are: side \(b = 6.3\), angle \(A = 24^\circ\), and angle \(C = 66^\circ\)

Step by step solution

01

Calculate the unknown side

First, solve for side \(b\). Apply Pythagoras' theorem which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as \(a^2 + b^2 = c^2\).Substitute the known values and solve for \(b\):\(b = \sqrt{a^2 - c^2} = \sqrt{7^2 - 3^2} = \sqrt{40}\). After rounding to the nearest tenth, \(b = 6.3\)
02

Calculate Angle A

Next, we will solve for angle \(A\). We can use the definition of sine in a right triangle, which is \(sin(A) = a/c\). Here, \(a\) is the side opposite to angle \(A\) and \(c\) is the hypotenuse. Substitute the known values and solve for \(A\):\(A = arcsin(a/c) = arcsin(3/7)\). After rounding to the nearest degree, \(A = 24^{\circ}\)
03

Calculate Angle C

Finally, we will solve for angle \(C\). In a triangle, the sum of all angles is \(180^{\circ}\), so we subtract the known angles from \(180^{\circ}\) to get \(C\): \(C = 180^{\circ} - A - B = 180^{\circ} - 24^{\circ} - 90^{\circ} = 66^{\circ}\)

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