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Chapter 5: Problem 28

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t\) \(\cos t,\) and \(\tan t.\) $$\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$

Short Answer

Expert verified
\( \sin t = \frac{\sqrt{3}}{2} \), \( \cos t = -\frac{1}{2} \), \( \tan t = -\sqrt{3} \).

Step by step solution

01

Understand the terminal point

The terminal point given is \( P \left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right) \). These correspond to \( x \) and \( y \) values on the unit circle, meaning \( x = \cos t \) and \( y = \sin t \). Therefore, \( \cos t = -\frac{1}{2} \) and \( \sin t = \frac{\sqrt{3}}{2} \).
02

Find \( \tan t \)

To find \( \tan t \), use the relationship \( \tan t = \frac{\sin t}{\cos t} \). Substituting the values \( \sin t = \frac{\sqrt{3}}{2} \) and \( \cos t = -\frac{1}{2} \), we have: \[ \tan t = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \]

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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, representing a circle with a radius of 1 centered at the origin of a coordinate plane. Using the unit circle, we can easily find the values of trigonometric functions for various angles.
Each point \(P(x, y)\) on the unit circle corresponds to the cosine and sine of an angle \(t\), where \(x = \cos t\) and \(y = \sin t\).
The unit circle allows us to see how angles correspond to points:
  • The angle is measured from the positive x-axis.
  • The point on the circumference gives \(x\) and \(y\) coordinates, which are the cosine and sine functions, respectively.
  • This illustrates the periodic nature of trigonometric functions.
Through this understanding, we can interpret trigonometric functions as direct reflections of geometric relationships on the circle.
Sine Function
The sine function, denoted as \(\sin t\), measures the y-coordinate of a point on the unit circle associated with an angle \(t\). In simpler terms, it reflects how far the point is from the x-axis vertically.
  • For the given point \((- rac{1}{2}, \frac{\sqrt{3}}{2})\), the sine value corresponds to the y-coordinate.
  • This makes \(\sin t = \frac{\sqrt{3}}{2}\).
  • The sine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
It is important to understand that as the angle \(t\) changes, the sine function value traces the y-values of the unit circle.
Cosine Function
The cosine function, \(\cos t\), represents the x-coordinate of a point on the unit circle associated with the angle \(t\). This function indicates how far the point is from the y-axis horizontally.
In our exercise:
  • The point \((x, y) = (-\frac{1}{2}, \frac{\sqrt{3}}{2})\) gives us \(\cos t = -\frac{1}{2}\).
  • Like the sine function, cosine is also periodic with a period of \(2\pi\).
  • The cosine function helps identify mirror and rotational symmetry on the unit circle.
Through analyzing these coordinates, one can deduce the angle's position relative to the unit circle's quadrants.
Tangent Function
The tangent function, \(\tan t\) is a bit different because it's a ratio of sine to cosine. Specifically, it is defined as: \(\tan t = \frac{\sin t}{\cos t}\).
It expresses the slope of the line that would reach from the origin to the point \(P(x, y)\) on the unit circle.
For our specific point:
  • Given \(\sin t = \frac{\sqrt{3}}{2}\) and \(\cos t = -\frac{1}{2}\), the tangent is:
  • \(\tan t = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}\)
  • The tangent function's period is \(\pi\), repeating every half circle interval.
This calculation of tangent helps us understand how steeply the angle diverges from the x-axis, enhancing our understanding of angle measures with respect to the unit circle.

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Most popular questions from this chapter

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From the information given, find the quadrant in which the terminal point determined by \(t\) lies. $$\csc t>0 \text { and } \sec t<0$$
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