Problem 49 - Algebra 2

Question

Use the definitions of trigonometric ratios in right \(\triangle A B C\) to verify each identity. \(\sec A=\frac{1}{\cos A}\)

Step-by-Step Solution

Answer

The given trigonometric identity \(sec A = \frac{1}{cos A}\) is indeed correct upon substitution of the definitions of the trigonometric functions into the identity and simplifying. The identity is therefore verified as true.

Step 1: Definition of Cos and Sec functions

Let us recall the definition of Trigonometric functions. The cosine function, cos A, in a right triangle is defined as the ratio of the length of the side adjacent to angle A (represented as AC) to the length of the hypotenuse (represented as BC). The secant function, sec A, is defined as the reciprocal of the cosine function i.e., \(sec A = \frac{1}{cos A}\). This is the identity that needs to be verified.

Step 2: Verify the Trigonometric Identity

Upon substitution of the definitions of the trigonometric functions into the identity, it would be evident that the identity holds true. (i.e., \(sec A = \frac{1}{cos A}\) becomes \(sec A = \frac{BC}{AC} = \frac{1}{\frac{AC}{BC}}\)). When we simplify, \(sec A = \frac{BC}{AC}\), hence verifying the identity as true.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees. In geometry, this is important because the 90-degree angle allows us to use specific trigonometric ratios and identities. In a right triangle, such as triangle ABC in the exercise, we usually label the sides as follows:

• The hypotenuse is the longest side and opposite the right angle.
• The opposite side is opposite to the angle we're focusing on, say angle A.
• The adjacent side is next to the angle A and not the hypotenuse.

These distinctions help us understand and use trigonometric functions more effectively. The structure of a right triangle is crucial because it provides the foundational base for applying concepts like cosine and secant.

Cosine

Cosine is one of the fundamental trigonometric functions and is typically represented as cos. In the context of a right triangle, specifically triangle ABC, cosine is defined based on angle A. It is the ratio of the length of the adjacent side to the hypotenuse.
The formula can be expressed as:
\[\cos A = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\]
In our example, the adjacent side is AC, and the hypotenuse is BC. Therefore, \(\cos A = \frac{AC}{BC}\).

Knowing how to calculate the cosine allows us to analyze the triangle’s geometry effectively, making predictions and solving for unknowns when the parameters change.

Secant

The secant function, often abbreviated as sec, is another trigonometric function intricately connected to cosine.

Secant is defined as the reciprocal of the cosine. Mathematically, this means:
\[\sec A = \frac{1}{\cos A}\]
In the context of triangle ABC, if \(\cos A = \frac{AC}{BC}\), then we substitute to find \(\sec A\):
\[\sec A = \frac{1}{\frac{AC}{BC}} = \frac{BC}{AC}\] The simplicity of secant makes it a handy tool when you need to reverse the action of cosine. Sometimes problems present information in a reciprocal form, and knowing secant helps untangle such challenges.

Trigonometric Ratios

Trigonometric ratios are relationships between the lengths of two sides in a right triangle. They are useful in solving geometric and real-world problems that involve angles and distances.

In a right triangle, the primary trigonometric ratios are:

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

These ratios allow you to link angles and distances in a comprehensible way.
When you understand these basic ratios, you can navigate through more complex problems in trigonometry, like verifying identities such as \(\sec A = \frac{1}{\cos A}\). Such knowledge is not only foundational but also expands understanding in fields requiring spatial and angular comprehension, like physics and engineering.