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We are given a right triangle \( \triangle IJK \), where \( \angle J = 90^\circ \), \( \overline{IJ} = 79 \), \( \overline{JK} = 46 \), and we are tasked with solving for the angle \( x^\circ \), which is \( \angle I \).

Since this is a right triangle, we can use trigonometric ratios to find the angle \( x^\circ \).

1. The adjacent side to \( \angle I \) is \( \overline{JK} = 46 \).
2. The opposite side to \( \angle I \) is \( \overline{IJ} = 79 \).

We can use the tangent function, which is defined as:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substituting the known values:

\[
\tan(x) = \frac{79}{46}
\]

Now, solving for \( x \):

\[
x = \tan^{-1}\left(\frac{79}{46}\right)
\]

Let me compute that.The angle \( x \) is approximately \( 59.8^\circ \) when rounded to the nearest tenth of a degree.

Would you like further details or have any additional questions?

Here are 5 related questions for practice:
1. How would you calculate the hypotenuse of triangle \( IJK \) using the given side lengths?
2. What is the measure of \( \angle K \) in triangle \( IJK \)?
3. If the length of side \( IJ \) were changed, how would it affect the value of \( x \)?
4. How can the sine and cosine of \( x \) be computed?
5. How do we confirm the sum of the angles in triangle \( IJK \) equals \( 180^\circ \)?

**Tip**: When dealing with right triangles, knowing two sides is enough to determine all the angles using trigonometric ratios.

Solve for x. Round to the nearest tenth of a degree, if necessary. Triangle IJK is given, where angle J is 90°, and the sides opposite and adjacent to angle I have lengths 79 and 46 respectively.

Learn how to solve for angle x in triangle IJK using the tangent function and trigonometric ratios. This problem involves using the lengths of two sides to calculate the angle in a right triangle.

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