Math Problem Statement

Solve the given triangle by finding all missing sides and missing angles. (Round your answers to three decimal places.)

Solution

The problem requires solving a right triangle by finding all missing sides and angles. The triangle has the following given information:

  • Angle A=39A = 39^\circ
  • Angle C=51C = 51^\circ (since the two angles are complementary in a right triangle)
  • Side AB=19AB = 19 (adjacent side to AA and opposite to CC).

We need to find:

  1. Hypotenuse (ACAC).
  2. Opposite side to AA (BCBC).

Step-by-Step Solution:

  1. Find BCBC (opposite side to AA): Using the tangent function: tan(A)=oppositeadjacent=BCAB\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} Rearrange to find BCBC: BC=ABtan(A)=19tan(39)BC = AB \cdot \tan(A) = 19 \cdot \tan(39^\circ) BC=190.809815.386BC = 19 \cdot 0.8098 \approx 15.386

  2. Find ACAC (the hypotenuse): Using the cosine function: cos(A)=adjacenthypotenuse=ABAC\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} Rearrange to find ACAC: AC=ABcos(A)=19cos(39)AC = \frac{AB}{\cos(A)} = \frac{19}{\cos(39^\circ)} AC=190.777124.445AC = \frac{19}{0.7771} \approx 24.445

  3. Verify the angles: Since this is a right triangle, the angles add up to 9090^\circ: A+C=39+51=90A + C = 39^\circ + 51^\circ = 90^\circ

Final Answer:

  • BC=15.386BC = 15.386
  • AC=24.445AC = 24.445
  • Angles are correct (A=39A = 39^\circ, C=51C = 51^\circ).

Let me know if you’d like a more detailed explanation for any step!


Related Questions:

  1. How can you verify the sides using the Pythagorean theorem?
  2. What are some real-life applications of solving right triangles?
  3. How would the solution change if AA were 4545^\circ?
  4. Why is the tangent function helpful for finding the opposite side?
  5. How does the cosine function relate to the hypotenuse?

Tip: Always double-check trigonometric values using a calculator or table for accuracy!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle
Angles

Formulas

tan(A) = opposite/adjacent
cos(A) = adjacent/hypotenuse

Theorems

Pythagorean Theorem
Trigonometric Ratios

Suitable Grade Level

Grades 10-12