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Chapter 3: Problem 10

Evaluate each statement given in exercises \(6-10\). $$ \sin \left(\frac{3 \pi}{2}\right) $$

Short Answer

Expert verified
\( \sin \left(\frac{3\pi}{2}\right) = -1 \).

Step by step solution

01

Understanding the Expression

We are given the expression \( \sin \left(\frac{3\pi}{2}\right) \). This asks us to find the sine of \( \frac{3\pi}{2} \), which is an angle measured in radians.
02

Identify the Angle on the Unit Circle

The angle \( \frac{3\pi}{2} \) in radians is equal to 270 degrees. On the unit circle, this angle corresponds to the negative y-axis.
03

Sine Value of the Angle

On the unit circle, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. For \( \frac{3\pi}{2} \), this point is (0, -1).
04

Conclusion

Since the sine of an angle is the y-coordinate of the corresponding point on the unit circle, \( \sin \left(\frac{3\pi}{2}\right) = -1 \).

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Key Concepts

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

Unit Circle
The unit circle is a fundamental concept in trigonometry, providing a geometric interpretation of trigonometric functions. It is a circle with a radius of one unit centered at the origin of a coordinate plane.
  • Every point on the unit circle is determined by an angle measured from the positive x-axis. This angle can be measured in either degrees or radians.
  • A point on the unit circle can be expressed in terms of coordinates \((x, y)\), which correspond to \((\cos(\theta), \sin(\theta))\) for an angle \(\theta\).
  • The x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine of the angle.
The unit circle allows us to visualize the periodic nature of the sine and cosine functions. Angles exceeding 360 degrees or 2\(\pi\) radians wrap around the circle, returning to their starting point. This wrapping characteristic makes the unit circle an essential tool for understanding trigonometric identities and relationships.
Angle Measurement
Angle measurement is crucial in trigonometry as it tells us how much we have rotated around a circle. There are two main units for angle measurement: degrees and radians.
  • Degrees divide a circle into 360 equal parts. Each part is one degree \((1^\circ)\).
  • Radians, however, relate the angle to the arc length of a circle. One radian is the angle formed when the arc length equals the radius of the circle.
  • The full circle, therefore, measures \(2\pi\) radians or 360 degrees, making conversion between the two units possible through the equivalence \(\pi\) radians = 180 degrees.
Understanding radians is essential in higher mathematics and physics as they integrate seamlessly into calculus. The expression \(\frac{3\pi}{2}\) radians, for example, corresponds to 270 degrees, taking us three-quarters of the way around the unit circle.
Trigonometric Functions
Trigonometric functions describe relationships between the angles and sides of triangles. They extend these relationships to the unit circle and are pivotal in various scientific fields.
  • The primary trigonometric functions are sine \((\sin)\), cosine \((\cos)\), and tangent \((\tan)\).
  • Sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. On the unit circle, this translates to the y-coordinate of the point associated with the angle.
  • Cosine corresponds to the x-coordinate of the same point, and tangent is the ratio of sine to cosine.
These functions are periodic, with sine and cosine repeating every full rotation of the circle. In practical problems like determining \(\sin \left(\frac{3\pi}{2}\right)\), the unit circle shows us that at this angle, the sine is \(-1\), as it lies on the negative y-axis at the point \( (0, -1) \). This consistent relationship allows us to deduce trigonometric function values effortlessly.

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Most popular questions from this chapter

Determine if it is possible to solve the statement for the given variable. If it is possible, solve but do not simplify your answer(s). If it is not possible, explain why. \(x y^{2}-x y=5 y-3 x\) for \(x\)

Solve for \(x .\) Be sure to list all possible values of \(x\). $$ 5 x^{2}+2 x=-1 $$

Determine if it is possible to solve the statement for the given variable. If it is possible, solve but do not simplify your answer(s). If it is not possible, explain why. \(2 a^{2} b c^{3}+3 a b c^{2}+4 a^{2} c^{2}-3 b=4 c\) for \(a\)

Determine the type of statement in terms of the given variable. \(x^{3} y+2 x^{2} y z-6 x z^{2}=y z^{2}-10\) in terms of \(z\)

Sketch the region bounded by the given functions and determine all intersection points. $$ y=x^{2} \text { and } y=x+2 $$
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