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Solve for \( x \).

Given a right triangle with a hypotenuse of 12 cm and an angle of 62 degrees, find the length of the side opposite the 62-degree angle (\( x \)).

\( x = [?] \) cm

Round to the nearest hundredth.

Answer :

To find the length of the side opposite the 62-degree angle in a right triangle with a hypotenuse of 12 cm, we can use the sine function from trigonometry.

The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. Mathematically, this is expressed as:

[tex]\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}[/tex]

Where:

  • [tex]\theta[/tex] is the given angle (62 degrees in this case).
  • The "opposite" side is the side we are trying to find (denoted as [tex]x[/tex]).
  • The "hypotenuse" is the longest side of the right triangle (given as 12 cm).

Let's plug in the values to solve for [tex]x[/tex]:

[tex]\sin(62^\circ) = \frac{x}{12}[/tex]

The sine of 62 degrees can be found using a calculator:

[tex]\sin(62^\circ) \approx 0.8829[/tex]

Now substitute this value back into the equation:

[tex]0.8829 = \frac{x}{12}[/tex]

To solve for [tex]x[/tex], multiply both sides by 12:

[tex]x = 12 \times 0.8829[/tex]

[tex]x \approx 10.59[/tex]

Therefore, the length of the side opposite the 62-degree angle is approximately 10.59 cm, rounded to the nearest hundredth.

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