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Chapter 3: Problem 65

Give the exact value of each of the following: $$4 \sin \left(-\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The exact value is \(-2\sqrt{2}\).

Step by step solution

01

Understanding the Negative Angle

The sine function is odd, which means that \(\sin(-x) = -\sin(x)\). Therefore, \(\sin\left(-\frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right).\)
02

Determine the Exact Value of \( \sin\left(\frac{\pi}{4}\right)\)

From the unit circle or known values, we know that \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
03

Apply the Odd Function Property to Find \( \sin\left(-\frac{\pi}{4}\right)\)

Apply the odd function property from Step 1: \(\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).
04

Calculate \(4 \sin\left(-\frac{\pi}{4}\right)\)

Now, multiply the result from Step 3 by 4: \(4 \times -\frac{\sqrt{2}}{2} = -2\sqrt{2}\).

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Key Concepts

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

Sine Function
The sine function is a vital trigonometric function used to relate angles to the unit circle. It specifically measures the vertical distance or height of a point on the unit circle corresponding to a certain angle from the positive x-axis. The sine of an angle is represented as \( \sin(\theta) \), where \( \theta \) is the angle.
  • It ranges from -1 to 1.
  • It is periodic with a period of \(2\pi\).
  • The sine function is part of the fundamental trigonometric functions along with cosine and tangent.
For practical learning, visualize the sine function as tracing a smooth curve, like waves, when plotted on a graph. This illustration helps in understanding how the sine function behaves as the angle changes from 0 to \(2\pi\) and beyond. This cyclical nature is valuable when predicting the values in various mathematical applications.
Unit Circle
In trigonometry, the unit circle is a tool used to define trigonometric functions for all angles. It is a circle with a radius of 1 centered at the origin of a coordinate plane.
  • Each point on the unit circle represents \((\cos(\theta), \sin(\theta))\).
  • The angle \( \theta \) is measured from the positive x-axis, typically in radians or degrees.
  • The unit circle enables easy calculation of trigonometric functions' values at precise angles.
When learning about the unit circle, imagine plotting a small circle on graph paper, then marking points like \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \). Compute the sine and cosine at these points, which often yields renowned mathematical constants. Key points such as \(\frac{\pi}{4}\) result in practical sine and cosine values, such as \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). Understanding this geometric tool aids significantly in grasping the principles of trigonometry.
Negative Angle
Understanding angles is crucial in trigonometry, especially when dealing with negative angles. A negative angle is an angle measured clockwise rather than counterclockwise from the positive x-axis.
  • Negative angles loop clockwise around the unit circle.
  • In trigonometric terms, \( \sin(-\theta) \) refers to reflecting the value of \( \sin(\theta) \), which can be understood better through the odd function property.
Visualize a negative angle as starting from the positive x-axis and turning downwards. The angle \(-\frac{\pi}{4}\) is merely \(\frac{\pi}{4}\) rotated negatively. To find the sine of this angle, reflect the positive angle's value across the x-axis through the odd property explained next.
Odd Function Property
A defining trait of the sine function is that it is an odd function. This property means that the sine function has symmetry around the origin in its graph.
  • The mathematical definition: \( \sin(-x) = -\sin(x) \).
  • This reflects a point over the origin, indicating that reversing the angle will negate the sine value.
Understanding the odd function property is integral to resolving equations involving sine, especially with negative angles. When facing \( \sin(-\theta) \), this property simplifies the work by switching to \(-\sin(\theta)\). Knowing this symmetry aids in calculations, such as proving \( \sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \), as demonstrated in the solved exercise.

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Most popular questions from this chapter

Find the exact value of each of the following. $$ \cos 225^{\circ} $$

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Find the exact value of each of the following. $$ \cos 150^{\circ} $$

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