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Chapter 5: Problem 9

Fill in the blank to complete the trigonometric identity. \( \cos(-u) \)= ________

Short Answer

Expert verified
\(\cos(u)\)

Step by step solution

01

Understanding cosine function

Firstly, understand that cosine is an even function, meaning it's symmetric about the y-axis. This leads to the property that for any angle \( u \), \(\cos(u) = \cos(-u)\). This property is the logic behind the solution.
02

Apply the property of cosine function

Apply the above mentioned property to the given trigonometric identity. We're given \( \cos(-u) \) and we need to fill in the blank. Using the property of cosine function we know that \( \cos(-u) = \cos(u) \).

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Key Concepts

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

Cosine Function
The cosine function is a fundamental component in trigonometry, one of the branches of mathematics that deals with triangles and the relationships between their sides and angles. It's part of the three main trigonometric functions, which also include sine and tangent. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.

Here's how you can understand it better:
  • The cosine function is often denoted as \( \cos(\theta) \), where \( \theta \) is the angle.
  • On the unit circle, the cosine of an angle is the x-coordinate of the point where the angle's terminal side intersects the circle.
  • It has values ranging from -1 to 1.
  • These values repeat every \( 360^\circ \) or \( 2\pi \) radians, revealing its periodic nature.
Understanding the cosine function allows us to explore its properties more deeply, such as its classification as an even function.
Even Function
An even function is a special type of function in mathematics. It has a characteristic symmetry that makes it distinct. For any even function, the relationship \( f(x) = f(-x) \) holds true for all values of \( x \) within the domain of the function.

Why is this important with cosine? Well:
  • The cosine function is an even function. This is mathematically expressed as \( \cos(u) = \cos(-u) \).
  • The even property of the cosine function implies it mirrors itself across the y-axis.
  • This property allows us to simplify expressions and solve problems efficiently, as seen in the original exercise where \( \cos(-u) \) becomes \( \cos(u) \).
By recognizing cosine as an even function, we can quickly validate its behavior over negative angles, confirming its symmetry property.
Symmetry in Trigonometry
Symmetry plays a crucial role in understanding trigonometric functions. Many trigonometric functions exhibit certain symmetries that simplify complex calculations.

For cosine and other trigonometric functions:
  • Cosine exhibits y-axis symmetry, meaning it mirrors across this axis.
  • In the context of the unit circle, this means the x-coordinate remains unchanged if you reflect the angle over the y-axis.
  • This symmetry is a direct result of cosine being an even function.
  • Understanding this symmetry can aid in visualizing trigonometric problems and solutions more intuitively.
Overall, recognizing the symmetry inherent in trigonometric functions like cosine can streamline problem-solving and deepen comprehension of more complex trigonometric identities.

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Most popular questions from this chapter

In Exercises 81-90, use the product-to-sum formulas to write the product as a sum or difference. \( \sin (x + y) \cos(x - y) \)

In Exercises 77-80, find all solutions of the equation in the interval \( [0, 2\pi) \). Use a graphing utility to graph the equation and verify the solutions. \( \cos \dfrac{x}{2} - \sin x = 0 \)

In Exercises 91-98, use the sum-to-product formulas to write the sum or difference as a product. \( \cos \left(\theta + \dfrac{\pi}{2} \right) - \cos \left(\theta - \dfrac{\pi}{2} \right) \)

In Exercises 19-28, find the exact solutions of the equation in the interval \( [0, 2\pi) \). \( (\sin 2x + \cos 2x)^2 = 1 \)

In Exercises 59-66, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. \( 75^\circ \)
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