/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 The point is on the terminal sid... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 25

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-3,-\sqrt{7})$$

Short Answer

Expert verified
The exact values of the six trigonometric functions for the angle where the point (-3,-\sqrt{7}) lies on the terminal side are: Sin\(\theta\) = -\sqrt{7}/4, Cos\(\theta\) = -3/4, Tan\(\theta\) = \sqrt{7}/3, Cosec\(\theta\) = -4/\sqrt{7}, Sec\(\theta\) = -4/3, Cot\(\theta\) = 3/\sqrt{7}

Step by step solution

01

Calculate the radius \(r\)

First, calculate the radius or hypotenuse of the triangle formed by \(x\), \(y\), and \(r\). We can use the Pythagorean theorem to calculate \(r\). The formula is \(r = \sqrt{x^2+y^2}\). Here, \(x=-3\) and \(y=-\sqrt{7}\). So, \(r = \sqrt{(-3)^2+(-\sqrt{7})^2} = \sqrt{9+7} = \sqrt{16} = 4.\) So, \(r\) equals to 4.
02

Calculate the six trigonometric functions

Now use \(x\), \(y\), and \(r\) to calculate the exact value of the six trigonometric functions. In this case: \n1. Sin\(\theta\) = \(y/r = -\sqrt{7}/4\)\n2. Cos\(\theta\) = \(x/r = -3/4\)\n3. Tan\(\theta\) = \(y/x = \sqrt{7}/3\)\n4. Cosec\(\theta\) is the reciprocal of Sin\(\theta\) = \(-4/\sqrt{7}\)\n5. Sec\(\theta\) is the reciprocal of Cos\(\theta\) = \(-4/3\)\n6. Cot\(\theta\) is the reciprocal of Tan\(\theta\) = \(3/\sqrt{7}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in mathematics, especially in trigonometry. It relates the lengths of the sides of a right triangle. By the theorem, the square of the hypotenuse (the side opposite the right angle) is the sum of the squares of the other two sides.

For a point \(x, y\) on the terminal side in standard position, we use the coordinates as the legs of a right triangle. Therefore, the hypotenuse \(r\) can be calculated by \[ r = \sqrt{x^2 + y^2} \]. Here, the values given are \(x = -3\) and \(y = -\sqrt{7}\).

Plug into the formula:
  • \( r = \sqrt{(-3)^2 + (-\sqrt{7})^2} = \sqrt{9 + 7} = \sqrt{16} = 4 \)
This computation is crucial for determining other trigonometric functions.
Standard Position
Angles in trigonometry are often described in standard position, which means the angle's vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis.

This setup helps in defining trigonometric functions using coordinates. When the terminal side of the angle passes through a point like \(-3, -\sqrt{7}\), it forms a triangle with the x-axis and helps determine key values.

Moving to coordinates is a common strategy to visualize angles and makes calculating trigonometric functions possible using the point's location.
Reciprocal Identities
Reciprocal identities are essential in trigonometry, providing an easy way to find three additional functions: cosecant, secant, and cotangent.
  • Cosecant \( (\text{csc}) \) is the reciprocal of sine: \( \text{csc}(\theta) = \frac{1}{\sin(\theta)} \).
  • Secant \( (\text{sec}) \) is the reciprocal of cosine: \( \text{sec}(\theta) = \frac{1}{\cos(\theta)} \).
  • Cotangent \( (\text{cot}) \) is the reciprocal of tangent: \( \text{cot}(\theta) = \frac{1}{\tan(\theta)} \).
For the given point, we had calculated:
  • \( \text{csc}(\theta) = -\frac{4}{\sqrt{7}} \)
  • \( \text{sec}(\theta) = -\frac{4}{3} \)
  • \( \text{cot}(\theta) = \frac{3}{\sqrt{7}} \)
These identities offer a straightforward method to extend the range of functions without direct division.
Exact Values
Finding the exact values of trigonometric functions is crucial for precise calculations. These values stem from specific circle points or basic angles.

In our exercise, calculating the functions for the point \(-3, -\sqrt{7}\), with \(x, y,\) and \(r = 4\), we derived:
  • \( \sin(\theta) = -\frac{\sqrt{7}}{4} \)
  • \( \cos(\theta) = -\frac{3}{4} \)
  • \( \tan(\theta) = \frac{\sqrt{7}}{3} \)
Working with exact values helps avoid rounding errors, providing precise outputs necessary for mathematical proofs and applications.

Accurate calculation of these values assures consistent results in various trigonometric scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

True or False Determine whether the statement is true or false. Justify your answer. The tangent function can be used to model harmonic motion.

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptote of the graph. $$f(x)=e^{3 x}$$

A company that produces wakeboards forecasts monthly sales \(S\) during a two- year period to be $$S=2.7+0.142 t+2.2 \sin \left(\frac{\pi t}{6}-\frac{\pi}{2}\right)$$ where \(S\) is measured in hundreds of units and \(t\) is the time (in months), with \(t=1\) corresponding to January \(2014 .\) Estimate sales for each month. (a) January 2014 (b) February 2015 (c) May 2014 (d) June 2015

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sin \theta=\frac{5}{6}\)

Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials $$\sin x \approx x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \quad \text { and } \quad \cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}$$ where \(x\) is in radians. (a) Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare? (b) Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare? (c) Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How did the accuracy of the approximations change when an additional term was added?
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.