Math Problem Statement

Use the fact that the trigonometric functions are periodic to find the exact value of each expression: a) sec(420°), b) cot(17π/4).

Solution

Let's analyze the trigonometric problems provided in the image:

1. Problem Analysis

The task is to find the exact values of the trigonometric expressions using their periodic properties.

a) sec420\sec 420^\circ

The secant function, sec(x)\sec(x), is periodic with a period of 360360^\circ. To find sec(420)\sec(420^\circ), we can first reduce it within the range of 00^\circ to 360360^\circ:

420360=60420^\circ - 360^\circ = 60^\circ

Thus, sec(420)=sec(60)\sec(420^\circ) = \sec(60^\circ).

Now, we know that:

sec(60)=1cos(60)\sec(60^\circ) = \frac{1}{\cos(60^\circ)}

And:

cos(60)=12\cos(60^\circ) = \frac{1}{2}

So:

sec(60)=112=2\sec(60^\circ) = \frac{1}{\frac{1}{2}} = 2

Therefore:

sec(420)=2\sec(420^\circ) = 2

b) cot(17π4)\cot \left( \frac{17\pi}{4} \right)

The cotangent function, cot(x)\cot(x), is periodic with a period of π\pi. To find cot(17π4)\cot \left( \frac{17\pi}{4} \right), we first reduce it within the range of 00 to π\pi:

17π44π=17π16π4=π4\frac{17\pi}{4} - 4\pi = \frac{17\pi - 16\pi}{4} = \frac{\pi}{4}

Thus:

cot(17π4)=cot(π4)\cot \left( \frac{17\pi}{4} \right) = \cot \left( \frac{\pi}{4} \right)

Now, we know:

cot(π4)=1\cot \left( \frac{\pi}{4} \right) = 1

Therefore:

cot(17π4)=1\cot \left( \frac{17\pi}{4} \right) = 1

Summary of Answers:

a) sec420=2\sec 420^\circ = 2

b) cot(17π4)=1\cot \left( \frac{17\pi}{4} \right) = 1

Would you like to go into more details about how these solutions were derived, or do you have any other questions?

Related Questions:

  1. What is the period of the tangent and cotangent functions?
  2. How do you find the exact values of trigonometric functions at specific angles?
  3. What are the co-function identities for sine and cosine?
  4. How do you determine the signs of trigonometric functions in different quadrants?
  5. How are trigonometric functions related to the unit circle?

Tip:

When working with trigonometric functions, always remember to reduce angles using their periods to find equivalent values within a standard range (e.g., 00^\circ to 360360^\circ or 00 to 2π2\pi).

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Periodic Functions
Secant
Cotangent

Formulas

sec(x) = 1/cos(x)
cot(x) = 1/tan(x)
Period of sec(x) = 360°
Period of cot(x) = π

Theorems

Periodic Properties of Trigonometric Functions
Exact Values of Trigonometric Functions

Suitable Grade Level

Grades 10-12