Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Complete the table with exact trigonometric function values. Do not use a calculator. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \\ \hline 45^{\circ} & & & 1 & 1 & & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

At 45°, \( \sin \) and \( \cos \) are \( \frac{\sqrt{2}}{2} \), \( \tan = \cot = 1 \), and \( \sec = \csc = \sqrt{2} \).

Step 1: Recall Basic Trigonometric Values

For angles like 45°, 30°, 60°, we can recall certain known values. For 45°, we know that the sine and cosine of the angle are equal, given by \( \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 2: Calculate Secant and Cosecant

The secant function is the reciprocal of the cosine, and the cosecant is the reciprocal of the sine. Therefore, \( \sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \) and \( \csc 45^{\circ} = \frac{1}{\sin 45^{\circ}} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \).

Step 3: Confirm Cotangent

The cotangent is the inverse of the tangent; since \( \tan 45^{\circ} = 1 \), we know that \( \cot 45^{\circ} = \frac{1}{1} = 1 \).

Step 4: Fill in the Table

Use these values to complete the table:
\( \begin{array}{|c|c|c|c|c|c|c|} \hline \theta & \sin \theta & \cos \theta & \tan \theta & \cot \theta & \sec \theta & \csc \theta \ \hline 45^{\circ} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 1 & 1 & \sqrt{2} & \sqrt{2} \ \hline \end{array} \)

Key Concepts

Sine and Cosine Values

The sine and cosine functions are fundamental in trigonometry. They relate the angles of a triangle to the lengths of its sides, specifically in right-angled triangles. For a given angle \( \theta \), the sine value is the ratio of the length of the opposite side to the hypotenuse, while the cosine value is the ratio of the length of the adjacent side to the hypotenuse.

One crucial aspect of sine and cosine values is how they behave for commonly used angles, such as \( 45^{\circ} \). At \( 45^{\circ} \), both the sine and cosine values are equal. This results from the fact that a \( 45^{\circ} \) angle in a right-angled triangle creates an isosceles right triangle, meaning the two legs are equal. Consequently, we have:

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

Keep in mind, these values are exact. They form the basis for the reciprocal functions discussed below.

Tangent and Cotangent

The tangent and cotangent functions are also essential in trigonometry. Tangent is the ratio of sine to cosine for any angle \( \theta \), expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). For \( 45^{\circ} \), this simplifies to 1, as the sine and cosine values are equal. Therefore:

• \( \tan 45^{\circ} = 1 \)

The cotangent is the reciprocal of the tangent, thus \( \cot \theta = \frac{1}{\tan \theta} \). For \( 45^{\circ} \), since the tangent is 1, the cotangent is also 1:

• \( \cot 45^{\circ} = 1 \)

Understanding tangent and cotangent helps link angles to circular motion concepts, which is pivotal in advanced trigonometric calculations.

Secant and Cosecant

Secant and cosecant are the reciprocal trigonometric functions of cosine and sine respectively. These functions often come into play when dealing with non-right triangles and in calculus applications.
For \( 45^{\circ} \), calculating the secant and cosecant involves taking the reciprocal of the corresponding cosine and sine values:

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

These exact values are not only important in solving triangles but also in understanding the behavior of waves and oscillations in physics and engineering applications. These functions help provide a complete picture of the relationships between the sides and angles of triangles, further enriching the study of trigonometric identities and integrals.