Problem 31 - Algebra 2

Question

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\cot \theta=\frac{5}{4}\)

Step-by-Step Solution

Answer

The five trigonometric ratios are: \( \sin \theta = \frac{4}{\sqrt{41}}\), \( \cos \theta = \frac{5}{\sqrt{41}}\), \( \tan \theta = \frac{4}{5}\), \( \sec \theta = \frac{\sqrt{41}}{5}\), and \( \csc \theta = \frac{\sqrt{41}}{4}\).

Step 1: Calculate the Hypotenuse

Using the Pythagorean theorem (\(a^2 + b^2 = c^2\)) where a and b are the given sides and c is the hypotenuse, we find \(c = \sqrt{a^2 + b^2}\). Substituting a = 5 and b = 4, we find \(c = \sqrt{5^2 + 4^2} = \sqrt{41}\). So, the hypotenuse is \(\sqrt{41}\)

Step 2: Find the Trigonometric Ratios

Substitute the calculated lengths of the sides to find all the trigonometric ratios. \- Sine of \(\theta\) (\(\sin \theta\)) is equal to \(\frac{Opposite Side}{Hypotenuse} = \frac{4}{\sqrt{41}}\). \- Cosine of \(\theta\) (\(\cos \theta\)) is equal to \(\frac{Adjacent Side}{Hypotenuse} = \frac{5}{\sqrt{41}}\). \- Tangent of \(\theta\) (\(\tan \theta\)) is equal to \(\frac{Opposite Side}{Adjacent Side} = \frac{4}{5}\). \- Secant of \(\theta\) (\(\sec \theta\)), which is the reciprocal of cosine, is equal to \(\frac{\sqrt{41}}{5}\). \- Cosecant of \(\theta\) (\(\csc \theta\)), which is the reciprocal of sine, is equal to \(\frac{\sqrt{41}}{4}\).

Step 3: Review and Check

Check all your calculations to ensure accuracy. It is important to verify that your values are correct and logically sound. In this case, all the trigonometric ratios obtained should be positive, since we are dealing with a right triangle where \(\theta\) is an acute angle.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right-angled triangles. This theorem helps us find the length of a triangle's sides when we know the lengths of two. The formula is \[a^2 + b^2 = c^2\]where:

• \(a\) and \(b\) are the lengths of the two legs
• \(c\) is the length of the hypotenuse, which is the side opposite the right angle

To apply this formula, simply square the lengths of the two legs, add them together, and then take the square root to find the hypotenuse. In our example, with sides 5 and 4, the hypotenuse is calculated as \(\sqrt{5^2 + 4^2} = \sqrt{41}\). This hypotenuse helps us find trigonometric ratios and solve for other unknowns.

Right Triangle

A right triangle is a special type of triangle with one 90-degree angle. This unique feature allows for the use of the Pythagorean Theorem and the trigonometric ratios that we commonly explore in trigonometry. Each right triangle has:

• One right angle
• Two other angles that are less than 90 degrees, known as acute angles
• Three sides: the hypotenuse (longest side), and the two legs

Right triangles are particularly useful because they help us understand angles and distances and solve real-world problems. In trigonometry, the relationships between the triangle's angles and sides are vital in calculations.

Acute Angle

An acute angle is any angle less than 90 degrees. In the context of a right triangle, the acute angles are the ones other than the 90-degree right angle. These angles are fundamental in determining the trigonometric ratios.Here's why they matter:

• Acute angles help calculate sine, cosine, and tangent ratios
• They are key to distinguishing the triangle's orientation and side lengths

Understanding acute angles aids in visualizing the triangle's layout and realizing which side is adjacent to or opposite the angle of interest. For instance, in our example, \(\theta\) is an acute angle, helping us identify the opposite side as 4 and the adjacent side as 5.

Trigonometry

Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles. While it might seem intimidating, its primary focus is using these relationships to solve problems.Key trigonometric functions include:

• Sine \(\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\)
• Cosine \(\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\)
• Tangent \(\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

These functions help describe a triangle's properties and calculate unknown angles or side lengths. In our example, they enable us to compute values like \(\sin \theta = \frac{4}{\sqrt{41}}\). Trigonometry is widely used in various fields like physics, engineering, and astronomy.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are extensions of the primary trigonometric functions. They offer alternate ways to express relationships between a triangle's angles and sides.These functions are:

• Cosecant \(\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite Side}}\)
• Secant \(\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent Side}}\)
• Cotangent \(\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent Side}}{\text{Opposite Side}}\)

These reciprocal functions can provide new insights or simplify computations. In our exercise, \(\csc \theta = \frac{\sqrt{41}}{4}\) shows the reciprocal nature of sine, offering a complete understanding of the triangle's geometry.