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Chapter 1: Problem 14

\(\sec 60^{\circ}\)

Short Answer

Expert verified
\( \sec 60^{\circ} = 2 \).

Step by step solution

01

Understanding Secant Function

The secant function, written as \( \sec \theta \), represents the reciprocal of the cosine function. That means \( \sec \theta = \frac{1}{\cos \theta} \). For this problem, \( \theta = 60^{\circ} \).
02

Recalling Cosine Value

Recall that \( \cos 60^{\circ} = \frac{1}{2} \). This is a commonly known trigonometric value.
03

Calculating Secant

To find \( \sec 60^{\circ} \), use the formula \( \sec 60^{\circ} = \frac{1}{\cos 60^{\circ}} \). Substitute the value we know: \( \sec 60^{\circ} = \frac{1}{\frac{1}{2}} \).
04

Simplifying the Expression

Simplify the expression: \( \frac{1}{\frac{1}{2}} = 2 \). Hence, \( \sec 60^{\circ} = 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function is one of the six fundamental trigonometric functions and plays a crucial role in various mathematical applications.
  • The notation for the secant function is \( \sec \theta \), and it is defined as the reciprocal of the cosine function.
  • This means that \( \sec \theta = \frac{1}{\cos \theta} \). In simpler terms, secant gives you the ratio of the hypotenuse length to the adjacent side length in a right triangle.
  • Secant is not commonly found in calculators, but being the reciprocal of cosine, it is just as useful.
Understanding secant helps in solving various trigonometric equations and can be particularly useful in calculus and other advanced math fields.
Cosine Function
The cosine function, represented as \( \cos \theta \), is another key trigonometric function. It reflects the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
  • For an angle \( \theta \), its cosine value is calculated as \( \cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \).
  • In trigonometric tables and unit circle analyses, \( \cos 60^{\circ} \) is readily known to be \( \frac{1}{2} \).
This memorized value, \( \cos 60^{\circ} = \frac{1}{2} \), is crucial in both theoretical and practical applications, helping to compute other trigonometric functions like the secant.
Reciprocal Identities
Reciprocal identities are relationships within trigonometry that show how certain trigonometric functions relate to each other through reciprocals.
  • For instance, the reciprocal identity for cosine is secant, meaning \( \sec \theta = \frac{1}{\cos \theta} \).
  • Other reciprocal identities exist for sine, tangent, cotangent, and cosecant, each expressing one function in terms of another.
These identities simplify the solving of trigonometric equations and make it easier to understand how different functions interact, thus strengthening your trigonometry toolbox.

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Most popular questions from this chapter

Calculate \(\frac{\tan \left(25^{\circ} 10^{\prime} 15^{\prime \prime}\right)+\sec \left(46^{\circ} 14^{\prime} 26^{\prime \prime}\right)}{\csc \left(23^{\circ} 17^{\prime} 23^{\prime \prime}\right)}\)

Seven Wonders. Only one of the great Seven Wonders of the Ancient World is still standing-the Great Pyramid of Giza. Each of the base sides along the ground measures 230 meters. If a l-meter child casts a 90 -centimeter shadow at the same time the shadow of the pyramid extends 16 meters along the ground (beyond the base), approximately how tall is the Great Pyramid of Giza?

Calculate \(\sec 70^{\circ}\) the following two ways: a. Write down \(\cos 70^{\circ}\) (round to three decimal places), and then divide 1 by that number. Write the number to five decimal places. b. First find \(\cos 70^{\circ}\) and then find its reciprocal. Round the result to five decimal places.

Use a calculator to find \(\cos ^{-1}\left(\cos 17^{\circ}\right)\).

\(\sin 50^{\circ}=0.77\)
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