Problem 20 - Precalculus: Functions and Graphs

Question

Use a formula for negatives to find the exact value. $$\text { (a) } \cot \left(-225^{\circ}\right) \quad \text { (b) } \sec \left(-\frac{\pi}{4}\right) \quad \text { (c) } \csc \left(-45^{\circ}\right)$$

Step-by-Step Solution

Answer

(a) -1, (b) \(\sqrt{2}\), (c) \(-\sqrt{2}\)

Step 1: Recognize Trigonometric Function Parities

Identify whether the given trigonometric function is even or odd. Recall that cosine and secant are even functions while sine, tangent, cosecant, and cotangent are odd functions.

Step 2: Apply Function Parity to Find \( \cot(-225^{\circ}) \)

Since cotangent is an odd function, \( \cot(-\theta) = -\cot(\theta) \). Thus, \( \cot(-225^{\circ}) = -\cot(225^{\circ}) \).

Step 3: Calculate \( \cot(225^{\circ}) \)

225° is in the third quadrant where both sine and cosine are negative, so \( \cot(225^{\circ}) = \frac{\cos 225^{\circ}}{\sin 225^{\circ}} = \frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1 \).

Step 4: Calculate Exact Value of \( \cot(-225^{\circ}) \)

Given that \( \cot(-225^{\circ}) = -\cot(225^{\circ}) \), and \( \cot(225^{\circ}) = 1 \), we have \( \cot(-225^{\circ}) = -1 \).

Step 5: Apply Function Parity to Find \( \sec \left(-\frac{\pi}{4}\right) \)

Secant is an even function, meaning \( \sec(-\theta) = \sec(\theta) \). Therefore, \( \sec \left(-\frac{\pi}{4}\right) = \sec \left(\frac{\pi}{4}\right) \).

Step 6: Calculate \( \sec \left(\frac{\pi}{4}\right) \)

The angle \( \frac{\pi}{4} \) corresponds to 45°, where the cosine of the angle is \( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \). Thus, \( \sec \left(\frac{\pi}{4}\right) = \frac{1}{\cos \left(\frac{\pi}{4}\right)} = \sqrt{2} \).

Step 7: Calculate Exact Value of \( \sec \left(-\frac{\pi}{4}\right) \)

As \( \sec \left(-\frac{\pi}{4}\right) = \sec \left(\frac{\pi}{4}\right) \), and \( \sec \left(\frac{\pi}{4}\right) = \sqrt{2} \), we have \( \sec \left(-\frac{\pi}{4}\right) = \sqrt{2} \).

Step 8: Apply Function Parity to Find \( \csc(-45^{\circ}) \)

Cosecant is an odd function, so \( \csc(-\theta) = -\csc(\theta) \). Therefore, \( \csc(-45^{\circ}) = -\csc(45^{\circ}) \).

Step 9: Calculate \( \csc(45^{\circ}) \)

At 45°, \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \). Thus, \( \csc(45^{\circ}) = \frac{1}{\sin(45^{\circ})} = \sqrt{2} \).

Step 10: Calculate Exact Value of \( \csc(-45^{\circ}) \)

Given \( \csc(-45^{\circ}) = -\csc(45^{\circ}) \), and \( \csc(45^{\circ}) = \sqrt{2} \), we conclude \( \csc(-45^{\circ}) = -\sqrt{2} \).

Key Concepts

Even and Odd Functions

Understanding the difference between even and odd functions is essential in trigonometry. These properties help us manipulate and find the exact values of functions quickly. An **even function** is symmetric about the y-axis. This means that for any function \( f \), if \( f(x) = f(-x) \), then it's even. Examples include cosine and secant. When you see these, remember they don't change with negation of the angle:

• \( \cos(-\theta) = \cos(\theta) \)
• \( \sec(-\theta) = \sec(\theta) \)

A function is **odd** when it has rotational symmetry about the origin. That is, \( f(x) = -f(-x) \). This is the case for sine, tangent, cosecant, and cotangent:

• \( \sin(-\theta) = -\sin(\theta) \)
• \( \cot(-\theta) = -\cot(\theta) \)

These parities simplify evaluations by turning angles into their positive counterparts or simply flipping the value's sign.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, describing relationships in triangles and cycles. The primary functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each has its own role in describing the dimensions of right-angled triangles.

• **Sine** describes the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine** is the ratio of the adjacent side to the hypotenuse.
• **Tangent** represents the ratio of the opposite side to the adjacent side.

From these arise the secondary functions: cotangent (\( \cot \)), secant (\( \sec \)), and cosecant (\( \csc \)). These are respectively the reciprocals of tangent, cosine, and sine. Understanding these relations is crucial in solving trigonometric equations.

Exact Values of Trigonometric Functions

Finding the exact value of trigonometric functions often involves specific angles, such as 30°, 45°, and 60°, or their radian equivalents. These angles have known sine, cosine, and tangent values. Let's look into two steps:

• **Step 1: Use known Angles** - Knowing that \( \sin(45°) = \cos(45°) = \frac{\sqrt{2}}{2} \) can simplify finding cosecant, secant, and cotangent for these angles.
• **Step 2: Apply identities** - Each function's reciprocal, as in \( \sec(45°) = \frac{1}{\cos(45°)} \), gives us \( \sqrt{2} \). These specific values help in calculating other angular measures too.

This knowledge accelerates solving any trigonometric problems without relying entirely on a calculator.