Question

3 practice 3
priya says, \i know everything about a right triangle with a 30 degree angle and a hypotenuse with length 1 cm. here, look.\

• the other angle is 60 degrees.
• the leg adjacent to the 30 degree angle is 0.866 cm long.
• the side opposite the 30 degree angle is 0.5 cm long.

han asks, \what would happen if a right triangle with a 30 degree angle has a hypotenuse that is 2 cm instead?\
help them find the missing angles and side lengths in the new triangle.
type your answers in the boxes.
missing angles: ____° and ____°
missing side lengths: ____ cm and ____ cm
explain or show your reasoning.
text input area and drawing area omitted

Explanation:

Step1: Find the missing angle

In a right - triangle, the sum of angles is \(180^{\circ}\). One angle is \(90^{\circ}\) (right angle) and one is \(30^{\circ}\). Let the missing angle be \(x\).

$$90 + 30+x=180$$

$$x = 180-(90 + 30)=60^{\circ}$$

Step2: Find the side opposite the \(30^{\circ}\) angle

We know the formula \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\theta = 30^{\circ}\) and hypotenuse \(c = 2\) cm.
\(\sin30^{\circ}=\frac{a}{2}\), since \(\sin30^{\circ}=\frac{1}{2}\), then \(a = 2\times\sin30^{\circ}=2\times\frac{1}{2}=1\) cm.

Step3: Find the side adjacent to the \(30^{\circ}\) angle

We know the formula \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta = 30^{\circ}\) and hypotenuse \(c = 2\) cm.
\(\cos30^{\circ}=\frac{b}{2}\), since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\), then \(b = 2\times\cos30^{\circ}=2\times0.866 = 1.732\) cm.

Answer:

Missing angles: \(60\) and \(90\)
Missing side lengths: \(1\) and \(1.732\)