Math Problem Statement

solve for length x using trigonometric ratio

Solution

To solve for the length xx in the right triangle using trigonometric ratios, we can use the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

In this triangle:

  • The angle given is 4040^\circ.
  • The side opposite this angle is the side with length xx.
  • The side adjacent to this angle is the side with length 77.

The tangent function is given by:

tan(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

Substituting the given values:

tan(40)=x7\tan(40^\circ) = \frac{x}{7}

Now, solve for xx:

x=7×tan(40)x = 7 \times \tan(40^\circ)

Let's calculate this value.

tan(40)0.8391\tan(40^\circ) \approx 0.8391

x7×0.83915.8737x \approx 7 \times 0.8391 \approx 5.8737

Thus, the length xx is approximately 5.875.87 units.

Would you like further details, or do you have any questions?

Here are some related questions you might consider:

  1. How can you use the sine function to find the hypotenuse in this triangle?
  2. What is the length of the hypotenuse using the sine function?
  3. Can we use the cosine function to find the length xx?
  4. How would you find the area of this right triangle?
  5. What is the value of the other non-right angle in this triangle?
  6. What would be the length of xx if the angle was 5050^\circ instead of 4040^\circ?
  7. How does the tangent function change as the angle increases?
  8. How can we verify the length xx using the Pythagorean theorem?

Tip: It's essential to double-check your trigonometric calculations, especially when working with angles, to ensure accuracy in your results.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right triangle

Formulas

Tangent function

Theorems

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Suitable Grade Level

Grades 9-12