Problem 3 - Algebra 2

Question

In \(\triangle G H I, \angle H\) is a right angle, \(G H=40,\) and \(\cos G=\frac{40}{41} .\) Draw a diagram and find each value in fraction and in decimal form. a. \(\sin G\) b. \(\sin I\) c. \(\cot G\) d. \(\csc G\) e. \(\cos I\) f. \(\sec I\)

Step-by-Step Solution

Answer

a) \(\sin G = \frac{9}{41}\) or 0.2195, b) \(\sin I = \frac{40}{41}\) or 0.9756, c) \(\cot G = \frac{40}{9}\) or 4.4444, d) \(\csc G = \frac{41}{9}\) or 4.5556, e) \(\cos I = \frac{9}{41}\) or 0.2195, f) \(\sec I = \frac{41}{9}\) or 4.5556.

Step 1: Calculate the length of side GI

From the Pythagorean theorem, we know that \(GI^2 = GH^2 + HI^2\). Since \(\angle H\) is a right angle, \(GH = 40\) is the base, and \(\cos G = \frac{GH}{GI} = \frac{40}{41}\), rearranging for GI gives \(GI = GH / \cos G = 41\).

Step 2: Calculate the length of side HI

Using the Pythagorean theorem again, we get \(HI^2 = GI^2 - GH^2 = 41^2 - 40^2 = 81\). Therefore, \(HI = \sqrt{81} = 9\).

Step 3: Calculate \(\sin G\)

\(\sin G = \frac{opposite}{hypotenuse} = \frac{HI}{GI} = \frac{9}{41}\) which in decimal form is approximately 0.2195.

Step 4: Calculate \(\sin I\)

\(\sin I = \frac{opposite}{hypotenuse} = \frac{GH}{GI} = \frac{40}{41}\) which in decimal form is approximately 0.9756.

Step 5: Calculate \(\cot G\)

\(\cot G = 1 / \tan G = \frac{adjacent}{opposite} = \frac{GH}{HI} = \frac{40}{9}\) which in decimal form is approximately 4.4444.

Step 6: Calculate \(\csc G\)

\(\csc G = 1 / \sin G = \frac{hypotenuse}{opposite} = \frac{GI}{HI} = \frac{41}{9}\) which in decimal form is approximately 4.5556.

Step 7: Calculate \(\cos I\)

\(\cos I = \frac{adjacent}{hypotenuse} = \frac{HI}{GI} = \frac{9}{41}\) which in decimal form is approximately 0.2195.

Step 8: Calculate \(\sec I\)

\(\sec I = 1 / \cos I = \frac{hypotenuse}{adjacent} = \frac{GI}{HI} = \frac{41}{9}\) which in decimal form is approximately 4.5556.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This right angle is denoted by a square box in diagrams. In a right triangle:

• The side opposite the right angle is the longest side and is known as the hypotenuse.
• The other two sides are called the legs of the triangle.

For instance, in triangle \(\triangle GHI\) from the problem, \(\angle H\) is a right angle. Therefore, the side \(GI\) is the hypotenuse because it is opposite \(\angle H\). The sides \(GH\) and \(HI\) serve as the legs.
Understanding the structure of right triangles is the foundation for solving trigonometric problems, as many principles like the Pythagorean theorem and trigonometric ratios are based on the characteristics of these triangles.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that:
\[ a^2 + b^2 = c^2 \] where \(c\) is the hypotenuse, and \(a\) and \(b\) are the triangle's other two sides.
In the context of \(\triangle GHI\), we applied this theorem twice:

• First, to verify that the hypotenuse \(GI = 41\) using \(GH = 40\) and \(\cos G = \frac{40}{41}\).
• Second, to solve for the third side \(HI = 9\), with the calculation \(HI^2 = 41^2 - 40^2\).

These steps illustrate how the Pythagorean theorem helps determine any side of a right triangle if the other two sides are known. This is crucial when you're working with trigonometric problems.

Trigonometric Ratios

Trigonometric ratios are essential for understanding the relationships between angles and sides in right triangles. The primary trigonometric ratios are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These ratios are defined based on the lengths of a triangle's sides:

• \(\sin\theta = \frac{opposite}{hypotenuse}\)
• \(\cos\theta = \frac{adjacent}{hypotenuse}\)
• \(\tan\theta = \frac{opposite}{adjacent}\)

In solving the exercise, these ratios were key to finding values:

• \(\sin G = \frac{HI}{GI}\) for \(\angle G\), and similarly for \(\sin I, \cos I\), etc.

Trigonometric ratios allow you to find unknown side lengths or angles in right triangles when some measurements are already known. They are powerful tools that extend the basic Pythagorean knowledge.

Cosine

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. Written as \(\cos\theta\), it is fundamental in solving triangle problems.
In \(\triangle GHI\), we know \( \cos G = \frac{GH}{GI} = \frac{40}{41} \).
This tells us that angle \(G\) has the leg \(GH\), adjacent to it, taking a significant portion of the hypotenuse \(GI\). It's also useful for indirectly finding other trigonometric functions related to the same angle when cosine is known.
When you learn cosines, remember:

• It's directly tied to the adjacent and hypotenuse sides.
• If \(\cos\) is known, you can find other ratios like \(\sin\) or even apply identities like \(\sin^2\theta + \cos^2\theta = 1\).

These concepts are utilized throughout trigonometry and solve real-life problems involving angles and distances.

Sine

The sine function is another key trigonometric ratio, often represented as \(\sin\theta\). It is the ratio of the side opposite to the angle to the hypotenuse.
For instance, in \(\triangle GHI\), we found \(\sin G = \frac{HI}{GI} = \frac{9}{41}\).
Here's what the sine function entails:

• \(\sin\theta\) gives a measure of how steep the triangle is from the perspective of that angle.
• If \(\sin\) of an angle is known, other relationships, such as \(\cos\), can be found using trigonometric identities.

Sine and cosine often work hand in hand. Understanding these concepts can let us explore problems involving waves, pendulums, and even sound. Being comfortable with sine helps you solve complex trigonometric problems with ease.