Math Problem Statement

Solve for x. Round to the nearest tenth of a degree, if necessary.

Solution

To solve for xx^\circ in the right triangle, we can use trigonometric ratios. Specifically, we know:

  • PQ=77PQ = 77 (the adjacent side to angle xx^\circ),
  • PO=67PO = 67 (the opposite side to angle xx^\circ).

Since we have both the opposite and adjacent sides, we use the tangent function:

tan(x)=oppositeadjacent=6777\tan(x^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{67}{77}

Now, we calculate xx^\circ:

x=tan1(6777)x = \tan^{-1}\left(\frac{67}{77}\right)

xtan1(0.8701)x \approx \tan^{-1}(0.8701)

Using a calculator:

x40.9x \approx 40.9^\circ

So, x40.9x \approx 40.9^\circ, rounded to the nearest tenth of a degree.

Would you like further details or have any questions?

Related Questions:

  1. How do we find an angle using sine or cosine in a right triangle?
  2. Can we solve for the other angles in this triangle?
  3. How can the Pythagorean theorem help in this problem?
  4. What is the sine of angle xx^\circ in this triangle?
  5. How can I use a calculator to find the inverse tangent function?

Tip:

Always ensure your calculator is set to degrees (not radians) when solving for angles in degrees!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles
Inverse Tangent Function

Formulas

tan(x) = opposite / adjacent
x = tan^{-1}(opposite / adjacent)

Theorems

Trigonometric Ratios

Suitable Grade Level

Grades 9-12