Problem 21 - Algebra 2 and Trigonometry

Question

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle R S T, r=15, s=18,\) and \(\mathrm{m} \angle T=90\)

Step-by-Step Solution

Answer

The hypotenuse is 23, \(\angle R = 50\degree\), and \(\angle S = 40\degree\).

Step 1: Identify Right Triangle Properties

Since angle \(T\) is 90 degrees, \(\triangle RST\) is a right triangle. We can use the Pythagorean theorem and trigonometric ratios (sine, cosine, tangent) to find the missing sides and angles.

Step 2: Use Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Here, we know \(r = 15\) and \(s = 18\). We need to find the hypotenuse \(t\). Let \(r = a\) and \(s = b\), so we have: \[ t^2 = r^2 + s^2 = 15^2 + 18^2 \]\[ t^2 = 225 + 324 \]\[ t^2 = 549 \]\[ t = \sqrt{549} \approx 23.4 \] Rounded to the nearest integer, \(t = 23\).

Step 3: Use Trigonometric Ratios to Find Angle R

Using the tangent function: \[ \tan(\angle R) = \frac{\text{opposite}}{\text{adjacent}} = \frac{s}{r} = \frac{18}{15} \]\[ \angle R = \tan^{-1} \left(\frac{18}{15}\right) \approx 50.2\degree \]Round to the nearest degree: \(\angle R = 50\degree\).

Step 4: Calculate Angle S

Since the angles in a triangle sum up to 180 degrees, and \(\angle T = 90\degree\), we have:\[ \angle R + \angle S + \angle T = 180\degree \]\[ \angle S = 180\degree - 90\degree - 50\degree = 40\degree \]

Step 5: Final Solution and Check

Now we have all the necessary parts:- \(r = 15\)- \(s = 18\)- \(t = 23\)- \(\angle R = 50\degree\)- \(\angle S = 40\degree\)- \(\angle T = 90\degree\)Verify by checking if \(\angle R + \angle S + \angle T = 180\degree\), which confirms our solution is consistent with triangle properties.

Key Concepts

Right Triangle

A right triangle is a triangle where one of the angles measures exactly 90 degrees. In our exercise, triangle RST has angle T equal to 90 degrees, making it a classic example of a right triangle. This type of triangle is especially important in trigonometry as it allows the use of specific theorems and ratios usually applicable only to right triangles.

Here are some key features of right triangles:

• The side opposite the right angle is called the hypotenuse, which is always the longest side.
• The other two sides are known as legs.

Understanding the properties of a right triangle helps you apply various mathematical tools, such as the Pythagorean theorem and trigonometric ratios. These tools simplify the process of solving for missing sides and angles.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used to find the relationships between the sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is expressed mathematically as:

\[c^2 = a^2 + b^2\]

Where \(c\) is the hypotenuse, and \(a\) and \(b\) are the triangle's legs. In the given exercise, we used it by substituting the known values of \(r = 15\) and \(s = 18\) to find the hypotenuse, \(t\). By calculating, we obtain:

• \(t^2 = 15^2 + 18^2 = 549\)
• \(t = \sqrt{549} \approx 23.4\)
• Rounded to the nearest integer, \(t = 23\)

Using the Pythagorean theorem can make solving right triangle problems straightforward and less time-consuming.

Trigonometric Ratios

Trigonometric ratios like sine, cosine, and tangent are crucial for solving right triangles and finding unknown angles and sides. These ratios relate the angles of a triangle to the lengths of its sides.

Here's how the tangent function was used in the problem:

• Tangent of an angle = Opposite side / Adjacent side
• In our triangle, \(\tan(\angle R) = \frac{s}{r} = \frac{18}{15}\)
• The angle \(\angle R\) is then calculated as the inverse tangent:
  \[\angle R = \tan^{-1}\left(\frac{18}{15}\right) \equiv 50\degree\]

Trigonometric ratios help provide more information about the triangles and are very effective for both solving and verifying problems involving angles and sides.

Angles of a Triangle

The sum of the angles in any triangle is always 180 degrees. This rule is essential in determining the remaining angles of a triangle when the measure of one or more angles is known.

For triangle RST:

• Given: \(\angle T = 90\degree\)
• We found: \(\angle R = 50\degree\)
• To find \(\angle S\), use the angle sum property:
  \[\angle S = 180\degree - \angle T - \angle R = 180\degree - 90\degree - 50\degree = 40\degree\]
• Verifying: \(\angle R + \angle S + \angle T = 180\degree\)

This simple rule helps ensure all calculations regarding the angles in any given triangle are correct and consistent.