Question

note: triangle may not be drawn to scale. suppose c = 7 and a = 70 degrees. find: a = b = b = degrees give all answers to at least one decimal place. give angles in degrees

Explanation:

Step1: Find angle B

In a right - triangle, the sum of the non - right angles is \(90^{\circ}\). Since it is a right - triangle (one angle is \(90^{\circ}\)) and \(\angle A = 70^{\circ}\), we use the formula \(\angle A+\angle B = 90^{\circ}\). So, \(\angle B=90^{\circ}-\angle A\). Substituting \(\angle A = 70^{\circ}\), we get \(\angle B = 90 - 70=20^{\circ}\).

Step2: Find side a (opposite to angle A)

We know that in a right - triangle, \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the hypotenuse \(c = 7\) and the side opposite to angle \(A\) is \(a\). So, \(\sin A=\frac{a}{c}\). Rearranging for \(a\), we have \(a = c\times\sin A\). Since \(A = 70^{\circ}\) and \(c = 7\), \(\sin(70^{\circ})\approx0.9397\). Then \(a=7\times\sin(70^{\circ})\approx7\times0.9397 = 6.5779\approx6.6\) (rounded to one decimal place).

Step3: Find side b (adjacent to angle A)

We know that in a right - triangle, \(\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}\). The side adjacent to angle \(A\) is \(b\) and the hypotenuse \(c = 7\). So, \(\cos A=\frac{b}{c}\). Rearranging for \(b\), we get \(b = c\times\cos A\). Since \(\cos(70^{\circ})\approx0.3420\) and \(c = 7\), \(b = 7\times\cos(70^{\circ})\approx7\times0.3420 = 2.394\approx2.4\) (rounded to one decimal place).

Answer:

\(a\approx6.6\), \(b\approx2.4\), \(B = 20.0\) degrees