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Chapter 6: Problem 15

Solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$A=65^{\circ}, B=65^{\circ}, c=6$$

Short Answer

Expert verified
The lengths of the sides of the triangle are a=5, b=4.6, c=2 and the measures of the angles are A=42 degrees, B=90 degrees and C=48 degrees.

Step by step solution

01

Applying Pythagoras theorem

Use Pythagoras theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. In this case, since angle B is 90 degrees, side c is the hypotenuse, so the theorem can be applied as follows: \(b^{2} = a^{2} - c^{2}\). Substituting given values, we get \(b^{2} = 5^{2} - 2^{2} = 21\). Therefore, b is \( \sqrt{21} = 4.6 \) when rounded to the nearest tenth.
02

Solving for angle 'A'

By definition of tangent, which is opposite over adjacent in a right triangle, we get \(\tan(A) = b/a\). Substituting given values: \(\tan(A) = 4.6/5\). Taking inverse tangent on both sides, \(\ A = \tan^{-1}(4.6/5) = 42^\circ \) when rounded to the nearest degree.
03

Solving for Other Angle

The sum of the angles in a triangle is \(180^\circ\). Thus we can calculate the unknown angle 'C' by using \(C = 180^\circ - A - B\). Substituting the known values, \( C = 180^\circ - 42^\circ - 90^\circ = 48^\circ \).

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