Math Problem Statement

Use a trigonometric ratio to find the length of x in the triangle. The angle is 28 degrees, and the adjacent side is 9 m. Give your answer to two decimal places.

Solution

To find the length of xx in this right triangle using trigonometric ratios, we can apply the tangent function because we have the angle 2828^\circ and the adjacent side (9 m). In a right triangle, the tangent of an angle is defined as:

tan(angle)=opposite sideadjacent side\tan(\text{angle}) = \frac{\text{opposite side}}{\text{adjacent side}}

For this problem:

  • The angle is 2828^\circ.
  • The adjacent side is 9 m.
  • The opposite side is xx.

Step-by-Step Solution

  1. Set up the equation using the tangent function: tan(28)=x9\tan(28^\circ) = \frac{x}{9}

  2. Solve for xx by multiplying both sides by 9: x=9×tan(28)x = 9 \times \tan(28^\circ)

  3. Calculate tan(28)\tan(28^\circ): tan(28)0.5317\tan(28^\circ) \approx 0.5317

  4. Substitute and solve: x9×0.5317x \approx 9 \times 0.5317 x4.79mx \approx 4.79 \, \text{m}

Answer

The length of xx is approximately 4.79 m.

Would you like more details on this solution or have any questions?


Here are 5 related questions for further practice:

  1. What trigonometric ratio would you use if you were given the opposite side and needed to find the hypotenuse?
  2. How would you calculate the adjacent side if you only knew the hypotenuse and the angle?
  3. If the angle were changed to 4545^\circ, what would the value of xx be?
  4. How would you find the length of the hypotenuse if you only had the adjacent side and the angle?
  5. What are the definitions of sine, cosine, and tangent in a right triangle?

Tip: Remember that tangent relates the opposite side to the adjacent side, which is especially useful in right triangles when you don’t have the hypotenuse.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle Ratios
Tangent Function

Formulas

tan(angle) = opposite / adjacent

Theorems

Trigonometric Ratios in Right Triangles

Suitable Grade Level

Grades 8-10