Question

2 multiple choice 1 point state whether the given measurements determine zero, one, or two triangles. a = 64°, a = 21, b = 23 one two zero

Explanation:

Step1: Use the Law of Sines

By the Law of Sines, \(\frac{\sin A}{a}=\frac{\sin B}{b}\). Substitute \(A = 64^{\circ}\), \(a = 21\), and \(b = 23\) into the formula: \(\sin B=\frac{b\sin A}{a}\).

$$ \sin B=\frac{23\sin64^{\circ}}{21} $$

$$ \sin B=\frac{23\times0.8988}{21}\approx\frac{20.6724}{21}\approx0.9844 $$

Step2: Analyze the value of \(\sin B\)

Since \(\sin B\approx0.9844\), and \(0\lt\sin B\lt1\). Also, \(a = 21\), \(b = 23\) (\(a\lt b\)) and \(A = 64^{\circ}\) (acute angle).
We know that if \(\sin B = k\) (\(0\lt k\lt1\)), and \(a\lt b\) with \(A\) acute, we check \(a\) and \(b\sin A\). Calculate \(b\sin A=23\sin64^{\circ}\approx23\times0.8988 = 20.6724\). Since \(a = 21\gt b\sin A\approx20.6724\), there are two possible values for \(B\) (one acute \(B_1=\sin^{- 1}(0.9844)\approx80.1^{\circ}\) and one obtuse \(B_2 = 180^{\circ}-80.1^{\circ}=99.9^{\circ}\)) that satisfy the triangle - angle sum formula \(A + B+C=180^{\circ}\).

Answer:

Two