/*! 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 91 Determine the signs of the trigo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 91

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. $$178^{\circ}$$

Short Answer

Expert verified
In the second quadrant, sine is positive, and cosine, tangent are negative for 178掳.

Step by step solution

01

Understand the quadrant

Determine which quadrant the angle is in by noting that it is in standard position and is measured from the positive x-axis. An angle of 178掳 is between 90掳 and 180掳, so it lies in the second quadrant.
02

Know the signs of trigonometric functions in the second quadrant

In the second quadrant, sine is positive, while cosine and tangent are negative. This is because in the second quadrant, the y-coordinates are positive and the x-coordinates are negative.
03

List the signs of each trigonometric function

Since the sine function corresponds to the y-coordinate, sine(178掳) is positive. The cosine function corresponds to the x-coordinate, so cosine(178掳) is negative. Tangent, being the ratio of sine to cosine, will also be negative since it is a positive number divided by a negative number.
04

Verify the other trigonometric functions

Cotangent, cosecant, and secant have signs based on their relationships with sine, cosine, and tangent. Cosecant (1/sine) remains positive. Secant (1/cosine) is negative. Cotangent (1/tangent) will be negative.

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.

angle in standard position
When studying trigonometry, it's crucial to understand what an angle in standard position is. This concept lays the foundation for determining the signs of trigonometric functions.
An angle is said to be in standard position if its vertex is at the origin of the coordinate system and its initial side lies along the positive x-axis.
As you measure the angle counterclockwise from the positive x-axis, you can determine the angle's terminal side.
For example, an angle of 178掳 means you rotate 178掳 from the positive x-axis.
This specific position helps to determine the angle's quadrant and, subsequently, the signs of its trigonometric functions.
quadrants in trigonometry
In trigonometry, the coordinate plane is divided into four quadrants:
  • First Quadrant (0掳 to 90掳)
  • Second Quadrant (90掳 to 180掳)
  • Third Quadrant (180掳 to 270掳)
  • Fourth Quadrant (270掳 to 360掳)

Each quadrant has unique characteristics that affect the signs of trigonometric functions.
For example, in the second quadrant, where angles range from 90掳 to 180掳, sine values are positive while cosine and tangent values are negative.
Our given angle of 178掳 lies within the second quadrant. This knowledge lets us easily determine which trigonometric functions are positive and which are negative.
signs of trigonometric functions
Understanding the signs of trigonometric functions in different quadrants is crucial for solving problems like this one. Let's break them down for each function:
  • Sine (sin): Positive in quadrants I and II, negative in quadrants III and IV.

  • Cosine (cos): Positive in quadrants I and IV, negative in quadrants II and III.

  • Tangent (tan): Positive in quadrants I and III, negative in quadrants II and IV.

Applying these to our angle of 178掳:
  • 蝉颈苍(178掳): Since it's in the second quadrant, the sine value is positive.

  • 肠辞蝉(178掳): In the second quadrant, the cosine value is negative.

  • 迟补苍(178掳): Tangent, being the ratio of sine to cosine, will be negative because a positive number (sine) divided by a negative number (cosine) is negative.

Additionally, the remaining trigonometric functions also depend on these values:
  • cosecant (1/sin): Positive, since sine is positive.

  • secant (1/cos): Negative, since cosine is negative.

  • cotangent (1/tan): Negative, since tangent is negative.

So, all trigonometric functions can be determined once you know the signs in each quadrant.

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

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable. $$234^{\circ}$$

As a consequence of Earth's rotation, celestial objects such as the moon and the stars appear to move across the sky, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 min in a direction perpendicular to Earth's axis.

Work each problem. In these exercises, assume the course of a plane or ship is on the indicated bearing. Two lighthouses are located on a north-south line. From lighthouse \(A\), the bearing of a ship 3742 m away is \(129^{\circ} 43^{\prime}\). From lighthouse \(B\), the bearing of the ship is \(39^{\circ} 43^{\prime}\). Find the distance between the lighthouses.

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\sin \theta, \text { given that } \csc \theta=1.42716321$$

Give an expression that generates all angles coterminal with each angle. Let n represent any integer. $$360^{\circ}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

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.