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Chapter 4: Problem 23

Evaluate (if possible) the six trigonometric functions at the real number. $$t=2 \pi / 3$$

Short Answer

Expert verified
The six trigonometric functions at the real number \( t = \frac{2\pi}{3} \) are: \( \sin \left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} \), \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \), \( \tan \left( \frac{2\pi}{3} \right) = -\sqrt{3} \), \( \cot \left( \frac{2\pi}{3} \right) = -\frac{1}{\sqrt{3}} \), \( \sec \left( \frac{2\pi}{3} \right) = -2 \), \( \csc \left( \frac{2\pi}{3} \right) = \frac{2}{\sqrt{3}} \)

Step by step solution

01

Find Sine value

Sine of \( t = \frac{2\pi}{3} \) can be found directly from the unit circle or the sine function graph. Since \( \frac{2\pi}{3} \) is in the second quadrant, where the sine function is positive, the sine value will be \( \frac{\sqrt{3}}{2} \). In other words, \( \sin \left( \frac{2\pi}{3} \right) = \frac{\sqrt{3}}{2} \)
02

Find Cosine value

Using a similar method, the cosine value can also be found. In the second quadrant, where \( t = \frac{2\pi}{3} \) lies, the cosine function is negative. So, the cosine value will be \(-\frac{1}{2}\) or \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \)
03

Find Tangent value

The tangent function of \( t \) is calculated as the ratio of sine and cosine. So, \( \tan \left( \frac{2\pi}{3} \right) = \frac{\sin \left( \frac{2\pi}{3} \right)}{\cos \left( \frac{2\pi}{3} \right)} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \)
04

Find Cotangent value

The cotangent function of \( t \) is the reciprocal of the tangent function. So, \( \cot \left( \frac{2\pi}{3} \right) = \frac{1}{\tan \left( \frac{2\pi}{3} \right)} = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} \)
05

Find Secant value

The secant value of \( t \) is the reciprocal of the cosine function. So, \( \sec \left( \frac{2\pi}{3} \right) = \frac{1}{\cos \left( \frac{2\pi}{3} \right)} = \frac{1}{-\frac{1}{2}} = -2 \)
06

Find Cosecant value

The cosecant function of \( t \) is the reciprocal of the sine function. So, \( \csc \left( \frac{2\pi}{3} \right) = \frac{1}{\sin \left( \frac{2\pi}{3} \right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \)

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Key Concepts

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

Sine
Sine is a fundamental trigonometric function that helps us find the vertical component of a point on the unit circle. When you have an angle, \( heta\), drawn from the positive x-axis, the sine tells us how far up or down the terminal point is on the unit circle.
  • The sine of an angle \( heta\) is written as \( ext{sin}( heta)\).
  • Sine values are positive in the first and second quadrants of the unit circle.
  • Using the Pythagorean identity, \( ext{sin}^2( heta) + ext{cos}^2( heta) = 1\), the sine value can be derived from known angles.
Understanding sine is pivotal, as it often appears in problems involving periodic phenomena like waves and oscillations. For the angle \( rac{2\pi}{3}\), it lies in the second quadrant. Thus, \( ext{sin}igg(\frac{2\pi}{3}\bigg)=\frac{\sqrt{3}}{2}\). This positive value is typical in situations when \( heta\) is in the second quadrant, illustrating sine's behavior within the unit circle.
Cosine
The cosine of an angle gives us its horizontal component on the unit circle. It tells how far left or right the terminal point is from the center.
  • Cosine for an angle \( heta\) is noted as \( ext{cos}( heta)\).
  • Cosine values are positive in the first and fourth quadrants.
  • Like sine, cosines are used with \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1\) identity to infer values.
For the angle \(\frac{2\pi}{3}\), which is in the second quadrant, the cosine is negative. \(\text{cos}\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\) captures this aspect of the cosine function in this quadrant. This sign change emphasizes cosine's unique trigonometric trait of altering with quadrants.
Tangent
The tangent of an angle relates the sine to the cosine, offering a ratio of vertical to horizontal positions. Tangent proves useful in fields like engineering and physics, where angles determine system states.
  • Tangent is expressed as \(\text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}\).
  • It determines the angle's slope or steepness.
  • Tangent is positive in the first and third quadrants.
For \(\frac{2\pi}{3}\), the tangent yields a dramatic value shift. Given that the sine is positive and the cosine negative here, the resulting tangent, \(\text{tan}\left(\frac{2\pi}{3}\right) = -\sqrt{3}\), turns negative, emphasizing how both functions govern its calculation.
Unit Circle
The unit circle is a pivotal tool in trigonometry, helping one visualize trigonometric functions on a plane. Imagine a circle with a radius of 1, centered at the origin of a coordinate system.
  • The x-coordinates represent cosine, whereas y-coordinates symbolize sine.
  • Each angle wraps around the circle, showing trigonometric values at various points.
  • Key angles, such as \(0, \pi/2, \pi, 3\pi/2\), are marked along the perimeter and each corresponds to significant sine and cosine values.
For \(\frac{2\pi}{3}\), it locates in the second quadrant, helping us determine that \(\text{sin}\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\text{cos}\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\). The unit circle assists us in translating angles into defined trigonometric values, crucial in varied mathematical contexts.

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Most popular questions from this chapter

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