Math Problem Statement

Find all sides and angles of the triangle

Solution

The given image shows a right triangle ABC\triangle ABC with:

  • B=62\angle B = 62^\circ
  • C=90\angle C = 90^\circ
  • AC=22.6AC = 22.6 cm

Step 1: Find A\angle A

Since ABC\triangle ABC is a right triangle, the sum of the angles in a triangle is 180180^\circ. Therefore, A\angle A can be found as:

A=180BC=1806290=28\angle A = 180^\circ - \angle B - \angle C = 180^\circ - 62^\circ - 90^\circ = 28^\circ

Step 2: Use trigonometric ratios to find ABAB and BCBC

  1. Finding ABAB (Hypotenuse):

Using the sine function: sin(62)=ACAB\sin(62^\circ) = \frac{AC}{AB} AB=ACsin(62)=22.6 cmsin(62)AB = \frac{AC}{\sin(62^\circ)} = \frac{22.6 \text{ cm}}{\sin(62^\circ)}

Calculating: sin(62)0.8829\sin(62^\circ) \approx 0.8829 AB=22.6 cm0.882925.6 cmAB = \frac{22.6 \text{ cm}}{0.8829} \approx 25.6 \text{ cm}

  1. Finding BCBC (Adjacent side to B\angle B):

Using the tangent function: tan(62)=ACBC\tan(62^\circ) = \frac{AC}{BC} BC=ACtan(62)=22.6 cmtan(62)BC = \frac{AC}{\tan(62^\circ)} = \frac{22.6 \text{ cm}}{\tan(62^\circ)}

Calculating: tan(62)1.8807\tan(62^\circ) \approx 1.8807 BC=22.6 cm1.880712.0 cmBC = \frac{22.6 \text{ cm}}{1.8807} \approx 12.0 \text{ cm}

Final Results:

  • A=28\angle A = 28^\circ
  • AB25.6AB \approx 25.6 cm
  • BC12.0BC \approx 12.0 cm

Would you like more details or have any questions?


Here are 5 related questions for further exploration:

  1. How would the triangle's sides change if B\angle B were larger?
  2. What is the area of ABC\triangle ABC based on the calculated sides?
  3. Can you use the Pythagorean theorem to verify the side lengths of ABC\triangle ABC?
  4. How would you find the length of side ABAB using the cosine of B\angle B?
  5. What happens to the trigonometric ratios if A\angle A were given instead of B\angle B?

Tip: When working with right triangles, always ensure the angle you reference corresponds to the correct side (opposite, adjacent, or hypotenuse) to use the right trigonometric function.

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Math Problem Analysis

Mathematical Concepts

Geometry
Trigonometry

Formulas

Sine function
Tangent function

Theorems

Sum of angles in a triangle

Suitable Grade Level

Grades 9-12