Math Problem Statement

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is in the correct mode.) sec 11𝜋/8

Solution

To evaluate sec(11π8)\sec\left(\frac{11\pi}{8}\right), we follow these steps:

  1. Recognize that secθ=1cosθ\sec \theta = \frac{1}{\cos \theta}. So we need to compute cos(11π8)\cos\left(\frac{11\pi}{8}\right) first.

  2. Ensure that the calculator is in radian mode since the angle is in terms of π\pi.

  3. Compute cos(11π8)\cos\left(\frac{11\pi}{8}\right).

  4. Take the reciprocal of that value to find sec(11π8)\sec\left(\frac{11\pi}{8}\right).

Now, let's calculate it:

sec(11π8)=1cos(11π8)\sec\left(\frac{11\pi}{8}\right) = \frac{1}{\cos\left(\frac{11\pi}{8}\right)}

Using a calculator:

cos(11π8)0.3827\cos\left(\frac{11\pi}{8}\right) \approx -0.3827

sec(11π8)=10.38272.6131\sec\left(\frac{11\pi}{8}\right) = \frac{1}{-0.3827} \approx -2.6131

So, the value of sec(11π8)\sec\left(\frac{11\pi}{8}\right) is approximately -2.6131.

Would you like further clarification or details?

Here are 5 related questions for further exploration:

  1. What is the sine of 11π8\frac{11\pi}{8}?
  2. How would you evaluate csc(11π8)\csc\left(\frac{11\pi}{8}\right)?
  3. Can you explain the difference between degrees and radians in trigonometry?
  4. What is the secant of 5π4\frac{5\pi}{4}, and how does it compare to 11π8\frac{11\pi}{8}?
  5. How do you determine the quadrant in which 11π8\frac{11\pi}{8} lies?

Tip: Always double-check that your calculator is in the correct mode (radians or degrees) when working with trigonometric functions to avoid incorrect results!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Radians
Secant Function
Cosine Function

Formulas

sec θ = 1 / cos θ
cos(θ) for angles in radians

Theorems

Reciprocal Identity: sec θ = 1 / cos θ
Unit Circle and Quadrant Analysis

Suitable Grade Level

Grades 9-12