Question

1. solve the following triangles. if needed, round your answers to one decimal pl

step 1: find angle b.
we know

step 2: find c.
we know

step 3: find b.
we know

Explanation:

Step 1: Find angle \( B \)

The sum of angles in a triangle is \( 180^{\circ} \). In a right - triangle (one angle is \( 90^{\circ} \)), if one of the non - right angles is \( 40^{\circ} \), then \( B=180^{\circ}-90^{\circ}-40^{\circ}=50^{\circ} \)

Step 2: Find \( c \)

We know that \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 40^{\circ}\), adjacent side to \(40^{\circ}\) is \(25\) cm (assuming the side of length \(25\) cm is adjacent to the \(40^{\circ}\) angle). So, \(\cos40^{\circ}=\frac{25}{c}\). Then \(c=\frac{25}{\cos40^{\circ}}\approx\frac{25}{0.766}\approx32.6\) cm

Step 3: Find \( b \)

We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 40^{\circ}\), adjacent side \(a = 25\) cm. So, \(\tan40^{\circ}=\frac{b}{25}\). Then \(b = 25\times\tan40^{\circ}\approx25\times0.839\approx21.0\) cm

Answer:

Angle \( B = 50^{\circ}\), \(c\approx32.6\) cm, \(b\approx21.0\) cm