91Ó°ÊÓ

A right triangle is a specific type of triangle that includes one angle measuring exactly 90 degrees. This right angle forms the cornerstone of many geometric principles, making right triangles crucial in both mathematics and real-world applications.

In right triangles, the side opposite the right angle is called the hypotenuse. This is always the longest side of the right triangle. The other two sides are referred to as the legs or simply sides of the triangle. These serve as the foundational elements when solving for unknown quantities within the triangle.

One important rule of right triangles is that the sum of the angles equals 180 degrees, just like all other triangles. However, since one angle is 90 degrees, the other two angles must sum to 90 degrees, often referred to as complementary angles. These properties set the groundwork for performing further calculations, such as solving for unknown side lengths and angle measures.

Calculating unknown angles in a right triangle is straightforward with the help of complementary angle principles. Since the triangle comprises one right angle (90 degrees), the other two angles (let's call them angle A and angle B) must add up to 90 degrees. This rule emerges from the property that the total sum of angles in any triangle is 180 degrees.

In the given problem, we are provided with angle A as 10 degrees 42 minutes. To find angle B, we subtract angle A from 90 degrees:

• Convert the given angle into a more manageable form if necessary (degrees and minutes to just degrees, for example). In this task, angle A is 10 degrees 42 minutes.
• Use the formula: \[ B = 90^\circ - A \]
• Where \( A \) has been given as \( 10^\circ 42^\prime \), subtract to find \( B = 90^\circ - 10^\circ 42^\prime = 79^\circ 18^\prime \)

This computation systematically uses the properties of triangle angles to ensure accuracy.

Trigonometric functions are essential tools in calculating unknown facets of triangles, particularly right triangles. They relate the angles of the triangle to the ratios of its sides. The three primary trigonometric functions are sine, cosine, and tangent.

• The sine function ( \( \sin \) ) relates an angle to the ratio of the opposite side over the hypotenuse: \[ \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \]
• The cosine function ( \( \cos \) ) relates an angle to the ratio of the adjacent side over the hypotenuse: \[ \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \]
• The tangent function ( \( \tan \) ) relates an angle to the ratio of the opposite side over the adjacent side: \[ \tan A = \frac{\text{opposite}}{\text{adjacent}} \]

These functions are integral in solving for unknown side lengths and angles in triangles when one angle and one side are known. In our case, the tangent function was used to find one of the side lengths, employing angle A and the given side length b.

Finding the hypotenuse in a right triangle often involves the application of trigonometric principles. The hypotenuse is the longest side opposite the right angle, and it holds key significance in various calculations.

In this exercise, we use the sine function to find the hypotenuse. Given the formula: \[ \sin A = \frac{a}{c} \] where \( a \) is one leg of the triangle and \( c \) is the hypotenuse, we rearrange it to solve for the hypotenuse as follows:

• \[ c = \frac{a}{\sin A} \]
• Find \( \sin(10^\circ 42^\prime) \) using a calculator.
• Substitute the values of \( a \) and \( \sin A \) to find \( c \). This highlights the hypotenuse's critical role in solving right triangles via trigonometric identities.