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:

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<reasoning>

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:
\[h \approx 10.0708\text{ cm}\]
</reasoning>

<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.
</answer>

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Answer:

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<reasoning>

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:
\[h \approx 10.0708\text{ cm}\]
</reasoning>

<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.
</answer>

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