Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the trigonometric function of the quadrant angle, if possible. $$\sec 0$$

Step-by-Step Solution

Answer

The value of \(\sec 0\) is 1.

Step 1: Understanding the function

The secant function is the reciprocal of the cosine function. In mathematical terms: \(\sec x = \frac{1}{\cos x}\).

Step 2: Apply the angle

Now we know that we have to apply 0 as our x, so the equation will be \(\sec 0 = \frac{1}{\cos 0}\).

Step 3: Find the cosine of the angle

The cosine of 0 degrees is 1, so \(\cos 0 = 1\).

Step 4: Compute the reciprocal

Now that we have cosine of the angle, the reciprocal is just 1 over this cosine value of 1. So, \(\frac{1}{\cos 0} = \frac{1}{1}\).

Step 5: Conclusion

After calculation, the secant of 0 end up being 1. Therefore, \(\sec 0 = 1\).

Key Concepts

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, which is a branch of mathematics concerned with the properties of triangles, especially right-angled triangles. The primary trigonometric functions are sine (\(\text{sin}\)), cosine (\(\text{cos}\)), and tangent (\(\text{tan}\)), and each is defined based on the ratios of sides of a right-angled triangle relative to one of its acute angles.

However, the scope of these functions expands far beyond triangles to describe relationships of angles and lengths on a unit circle, where the radius of the circle is 1. By examining a unit circle, you can determine the values of the trigonometric functions for any angle, not just those found in right-angled triangles. In our example, the secant function, which is one of the less commonly mentioned functions, is directly defined through the cosine function.

Quadrant Angles

In trigonometry, quadrant angles are the angles that fall exactly on the axes of the coordinate plane, dividing it into four quadrants. These specific angles are 0°, 90°, 180°, and 270°, which correspond to the starting point, and then anti-clockwise rotations to the top, the opposite side, and the bottom of the circle, respectively.

Understanding the properties of these quadrant angles is essential because they serve as the benchmarks from which we examine the behavior of trigonometric functions. Each quadrant associates different signs (+ or -) with trigonometric function values. For instance, in the first quadrant, where our example angle 0° lies, all trigonometric functions have positive values. This is why when we calculate \( \text{cos}(0) \), we obtain a positive 1, leading to \( \text{sec}(0) \) also being positive.

Reciprocal Trigonometric Identities

Reciprocal trigonometric identities are fundamental relationships within the study of trigonometry that describe how certain trigonometric functions can be expressed in terms of others through the use of reciprocals. The basic reciprocal identities include secant (\(\text{sec}\)), cosecant (\(\text{csc}\)), and cotangent (\(\text{cot}\)), which are respectively the reciprocals of cosine (\(\text{cos}\)), sine (\(\text{sin}\)), and tangent (\(\text{tan}\)).

In formulaic terms, these relationships are expressed as:\( \text{sec}(x) = \frac{1}{\text{cos}(x)} \), \( \text{csc}(x) = \frac{1}{\text{sin}(x)} \), and \( \text{cot}(x) = \frac{1}{\text{tan}(x)} \). Understanding these relationships allows us to simplify complex trigonometric expressions and solve trigonometric equations more efficiently. As reflected in our example, knowing that the cosine of 0 degrees is 1 informs us that the secant of 0 degrees is also 1, based on the reciprocal identity \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \).