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Chapter 1: Problem 12

In which quadrant(s) do sine and cosine have the opposite sign?

Short Answer

Expert verified
Quadrants II and IV

Step by step solution

01

- Understanding the Trigonometric Functions

Identify the properties of sine and cosine. The sine function represents the y-coordinate, and the cosine function represents the x-coordinate of a point on the unit circle.
02

- The Four Quadrants

Recall the four quadrants of the unit circle. In Quadrant I (0掳 to 90掳), both sine and cosine are positive. In Quadrant II (90掳 to 180掳), sine is positive, and cosine is negative. In Quadrant III (180掳 to 270掳), both sine and cosine are negative. In Quadrant IV (270掳 to 360掳), sine is negative, and cosine is positive.
03

- Identifying Opposite Signs

Determine the quadrants where sine and cosine have opposite signs. Quadrants II and IV are suitable. In Quadrant II, sine is positive and cosine is negative. In Quadrant IV, sine is negative and cosine is positive.

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Key Concepts

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

sine and cosine signs
In trigonometry, understanding the signs of sine and cosine is crucial.
These functions give us important information about angles and points on the unit circle.
The sine function represents the y-coordinate of a point on the unit circle, while the cosine function represents the x-coordinate.
The properties of these functions vary depending on the angle's location within the unit circle's four quadrants.
Recognizing when sine and cosine are positive or negative helps solve many trigonometric problems efficiently.
unit circle quadrants
The unit circle is divided into four quadrants, each with different sign conventions for sine and cosine.
Understanding these quadrants helps in identifying the behavior of trigonometric functions.
Here is a breakdown of the quadrants:
  • **Quadrant I (0掳 to 90掳)**: Both sine and cosine are positive.
  • **Quadrant II (90掳 to 180掳)**: Sine is positive, and cosine is negative.
  • **Quadrant III (180掳 to 270掳)**: Both sine and cosine are negative.
  • **Quadrant IV (270掳 to 360掳)**: Sine is negative, and cosine is positive.

Using this knowledge, we can determine which quadrants feature sine and cosine having opposite signs.
As seen, Quadrants II and IV have these properties.
trigonometric properties
Trigonometric functions have several fundamental properties.
Understanding these can significantly aid in solving problems like identifying signs in various quadrants.
Some essential properties include periodicity, symmetry, and sign conventions.
  • **Periodicity**: Sine and cosine functions are periodic with a period of 360掳 or 2蟺. This means that the functions repeat their values after one full circle.
  • **Symmetry**: Sine is an odd function, meaning sine(-x) = -sine(x). Cosine is an even function, meaning cosine(-x) = cosine(x).
  • **Sign Conventions**: As previously covered, the signs of sine and cosine change depending on the quadrant.

By understanding these properties, you can solve trigonometric problems more accurately and efficiently.

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Most popular questions from this chapter

Find all angles \(0^{\circ} \leq \theta<360^{\circ}\) which satisfy the given equation: $$\sin \theta=0$$

State in which quadrant or on which axis the given angle lies. $$313^{\circ}$$

State in which quadrant or on which axis the given angle lies. $$-313^{\circ}$$

Find the values of the other five trigonometric functions of the acute angle \(A\) given the indicated value of one of the functions. $$\cos A=\frac{2}{\sqrt{10}}$$

Prove that the hypotenuse is the longest side in every right triangle. (Hint: Is \(a^{2}+b^{2}>a^{2} ?\) )
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