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Chapter 5: Problem 29

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(\frac{3}{5}, \frac{4}{5}\right)$$

Short Answer

Expert verified
\(\sin t = \frac{4}{5}, \cos t = \frac{3}{5}, \tan t = \frac{4}{3}\).

Step by step solution

01

Identify the Terminal Point Coordinates

The terminal point given is \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\). From this, we identify \(x = \frac{3}{5}\) and \(y = \frac{4}{5}\).
02

Determine \(\sin t\)

The sine of the angle \(t\) is the ratio of the \(y\)-coordinate to the hypotenuse of the unit circle. Since we assume this point is on the unit circle, the hypotenuse is 1. Thus, \(\sin t = y = \frac{4}{5}\).
03

Determine \(\cos t\)

The cosine of the angle \(t\) is the ratio of the \(x\)-coordinate to the hypotenuse of the unit circle. Hence, \(\cos t = x = \frac{3}{5}\).
04

Determine \(\tan t\)

The tangent of the angle \(t\) is the ratio of the \(y\)-coordinate to the \(x\)-coordinate. Therefore, \(\tan t = \frac{y}{x} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\).

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Key Concepts

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

Understanding the Sine Function
The sine function is a critical concept in trigonometry that relates to the unit circle, which has a radius of one.
In the context of a point \(P(x, y)\) on the unit circle, \( \sin t \) is defined as the \(y\)-coordinate of that point.
In other words, if you have a point \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\), the sine of the angle \(t\) would simply be the \(y\)-value, which is \(\frac{4}{5}\).
  • The sine function indicates the vertical distance of the point \(P(x,y)\) from the origin.
  • It ranges between -1 and 1 as it represents coordinates on the unit circle.
Understanding this concept is foundational as it connects angles with linear distances in a circular manner.
Exploring the Cosine Function
Similarly to the sine function, the cosine function is also fundamentally linked to the unit circle.
The cosine of an angle \( t \) is the \(x \)-coordinate of the point on the unit circle determined by that angle.
In our example of the terminal point \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\), the \(x\)-value, or cosine, is \(\frac{3}{5}\).
  • Cosine represents how far along the horizontal axis the point \(P(x,y)\) is from the origin.
  • Just like the sine function, cosine values are bounded between -1 and 1.
The cosine function helps describe the horizontal aspect of an angle's terminal point on the unit circle.
Diving into the Tangent Function
The tangent function is a bit different from sine and cosine as it represents a ratio rather than just a coordinate.
The tangent of an angle \( t \) is calculated by dividing the \(y \)-coordinate by the \(x \)-coordinate at the terminal point on the unit circle.
For our terminal point \(P(x, y) = \left(\frac{3}{5}, \frac{4}{5}\right)\), the tangent is \(\tan t = \frac{4}{3}\).
  • The tangent function informs us of the steepness or inclination of the line connecting the origin to \(P(x,y)\).
  • While it doesn't have a fixed range as sine and cosine, it is undefined when \( x = 0 \) since you cannot divide by zero.
Thus, tangent provides a measure of directionality and slope for angles based on their position on the unit circle.

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Most popular questions from this chapter

Find the period and graph the function. $$y=\csc 2\left(x+\frac{\pi}{2}\right)$$

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\tan t, \cos t ; \quad\) Quadrant III

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

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). $$\left(\frac{\sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right)$$

Find the exact value of the expression, if it is defined. $$\cos ^{-1}\left(\cos \frac{5 \pi}{6}\right)$$
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