Question

follow the steps to find the area of the shaded region.

next, find the height (h) of the triangle inside the sector.

hint: use sohcahtoa

\\(h = ?\text{ cm}\\)

round to four decimal places.

Explanation:

Identify the given values in the right triangle

We are given a right triangle inside a circle sector with:

• Hypotenuse \(c = 14\text{ cm}\) (the radius of the circle)
• Angle opposite to the height \(h\), which is \(\theta = 46^\circ\)
• The height \(h\) is the side opposite to the \(46^\circ\) angle.

Apply the trigonometric ratio

To find the opposite side \(h\) when the hypotenuse is known, we use the sine function from SOHCAHTOA:

$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

Substituting the known values:

$$\sin(46^\circ) = \frac{h}{14}$$

Solve for the height h

Isolate \(h\) by multiplying both sides by \(14\):

$$h = 14 \cdot \sin(46^\circ)$$

Using a calculator to find \(\sin(46^\circ)\):

$$\sin(46^\circ) \approx 0.7193398$$

Calculate the value of \(h\):

$$h \approx 14 \cdot 0.7193398 \approx 10.070757$$

Round to four decimal places

Rounding \(10.070757\) to four decimal places gives:

$$h \approx 10.0708\text{ cm}$$

Answer:

Next, find the height (h) of the triangle inside the sector.

Hint: Use SOHCAHTOA

\(h =\) <blank>10.0708</blank> cm

Round to four decimal places.