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Chapter 8: Problem 26

Find \(t\) such that \(0 \leq t \leq \pi\) and \(t\) satisfies the stated condition. \(\cos t=\cos (-3 \pi / 4)\)

Short Answer

Expert verified
t = \(3\frac{\text{蟺}}}{4}\)

Step by step solution

01

Use the property of the cosine function

Recall that the cosine function is an even function, meaning \(\text{cos}(-x) = \text{cos}(x)\). Therefore, \(\text{cos}(-3\frac{\text{\text{蟺}}}{4}) = \text{cos}(3\frac{\text{\text{蟺}}}{4})\).
02

Find the principal value

Since we need \(0 \leq\ t \leq\ \pi\), we find the value of \(t\) such that \(t = \cos^{-1}(\text{cos}(3\frac{\text{蟺}}}{4}))\). The principal value of \(\text{cos^{-1}}(...)\) between \(0 \leq\ t \leq\ \pi\) is \(3\frac{\text{蟺}}}{4}\).

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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, denoted as \(\text{cos}(x)\), is a fundamental trigonometric function representing the x-coordinate of a point on the unit circle corresponding to a given angle \( x \). A few essential properties of the cosine function include:
- **Range**: Values of cosine lie between -1 and 1, i.e., \(-1 \leq \text{cos}(x) \leq 1\).
- **Periodicity**: It is periodic with period \(\text{2蟺}\), meaning \(\text{cos}(x) = \text{cos}(x + 2\text{蟺})\).
even function
An even function is symmetric about the y-axis, meaning its value at negative \( x \) is the same as its value at positive \( x \). For the cosine function, this property can be expressed as \(\text{cos}(-x) = \text{cos}(x)\). This is why \(\text{cos}(-3蟺/4)\) equals \(\text{cos}(3蟺/4)\). By using this symmetry, we can simplify solving trigonometric equations when \(-x\) is involved.
inverse trigonometric functions
Inverse trigonometric functions are used to determine angles given their trigonometric values. For the cosine function, the inverse is denoted as \( \text{cos}^{-1}(x) \). The range of \( \text{cos}^{-1}(x) \) is restricted to \(0 \leq t \leq 蟺\). This restriction ensures we obtain a unique principal value. In the original problem, knowing that \( t = \text{cos}^{-1}(\text{cos}(3蟺/4))\) within the specified interval confirms that the value of \( t \) is \(3蟺/4\).

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