Problem 37 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \left(\sec 45^{\circ}\right)^{2}-\left(\tan 45^{\circ}\right)^{2} $$

Step-by-Step Solution

Answer

The exact value is 1.

Step 1: Identify Trigonometric Values at 45 Degrees

First, we identify the trigonometric values at \( 45^{\circ} \). For \( 45^{\circ} \), we know: - \( \sec 45^{\circ} = \sqrt{2} \) - \( \tan 45^{\circ} = 1 \)

Step 2: Calculate Squared Values

Next, we square the trigonometric values identified in Step 1: - \( \sec^2 45^{\circ} = (\sqrt{2})^2 = 2 \) - \( \tan^2 45^{\circ} = 1^2 = 1 \)

Step 3: Subtract the Squared Values

With the squared values calculated, subtract \( \tan^2 45^{\circ} \) from \( \sec^2 45^{\circ} \): - \( \sec^2 45^{\circ} - \tan^2 45^{\circ} = 2 - 1 \)

Step 4: Calculate the Result

Complete the subtraction from Step 3 to find the answer: - \( 2 - 1 = 1 \)

Key Concepts

Understanding the Secant Function

The secant function, denoted as sec, is one of the reciprocal trigonometric functions. It's related closely to the cosine function. Specifically, the secant of an angle is the reciprocal of the cosine of that angle. In mathematical terms:

• \( \sec \theta = \frac{1}{\cos \theta} \)

For example, if you know the cosine of an angle, say \( \cos 45^{\circ} = \frac{1}{\sqrt{2}} \), then the secant can be found:

• \( \sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} = \sqrt{2} \)

The secant function has interesting properties and values at different angles, which can be especially helpful in solving problems involving identities, as we will see in exercises like the one above.

Exploring the Tangent Function

The tangent function is another fundamental trigonometric function, often abbreviated as tan. It plays a critical role in trigonometry and is defined in terms of the sine and cosine functions:

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

At certain key angles, the tangent function assumes values that are particularly significant and easy to remember. One such angle is \( 45^{\circ} \). At 45 degrees:

• \( \tan 45^{\circ} = 1 \)

This happens because both the sine and cosine of 45 degrees are equal, \( \frac{1}{\sqrt{2}} \), making their ratio equal to 1. Remembering the value of tangent at this angle helps in many trigonometric calculations and proofs.

Focusing on the 45 Degrees Angle

The 45 degrees angle is a special angle in trigonometry, offering symmetrical properties that simplify calculations. This angle is often encountered in situations involving right triangles where the two non-hypotenuse sides are equal. Some key trigonometric values at 45 degrees include:

• \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{1}{\sqrt{2}} \)
• \( \tan 45^{\circ} = 1 \)
• \( \sec 45^{\circ} = \sqrt{2} \)

These values are derived from the properties of an isosceles right triangle and are critical in solving problems involving the Pythagorean identity and others. Knowing these values can often offer a shortcut in calculations, as they did in the given exercise.

Recognizing Basic Trigonometric Values

Trigonometric identities and their values at key angles form the foundation of understanding trigonometry. Three basic trigonometric functions—sine, cosine, and tangent—have distinct and easy-to-remember values at 0, 30, 45, 60, and 90 degrees. For 45 degrees, these values are often highlighted due to their balance and symmetry. Considering the main functions:

• \( \sin 0^{\circ} = 0, \cos 0^{\circ} = 1 \)
• \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{1}{\sqrt{2}} \)
• \( \tan 45^{\circ} = 1 \)

Similarly, reciprocal functions such as secant and cosecant also derive straightforward values at these angles, for example, \( \sec 45^{\circ} = \sqrt{2} \). Having these values easily accessible can make solving exercises much more manageable, whether finding the solution manually or proving identities. They help you calculate quickly and verify your work with confidence.