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Chapter 6: Problem 78

Find the signs of the six trigonometric function values for the given angles. $$290^{\circ}$$

Short Answer

Expert verified
In the fourth quadrant, sine, tangent, cosecant, and cotangent are negative. Cosine and secant are positive.

Step by step solution

01

- Determine the Quadrant

First, find out in which quadrant the angle lies. The angle given is \(290^{\text{掳}}\). Since \(290^{\text{掳}}\) is between \(270^{\text{掳}}\) and \(360^{\text{掳}}\), the angle is in the fourth quadrant.
02

- Know the Signs in the Fourth Quadrant

In the fourth quadrant, the signs of the trigonometric functions are as follows: - Sine (sin) is negative - Cosine (cos) is positive - Tangent (tan) is negative - Cosecant (csc) is negative (since it is the reciprocal of sine) - Secant (sec) is positive (since it is the reciprocal of cosine) - Cotangent (cot) is negative (since it is the reciprocal of tangent)

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Key Concepts

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

trigonometric quadrants
Understanding in which quadrant an angle lies is essential for determining the signs of trigonometric functions. The coordinate plane is divided into four quadrants:

  • First Quadrant: Angles between 0掳 and 90掳
  • Second Quadrant: Angles between 90掳 and 180掳
  • Third Quadrant: Angles between 180掳 and 270掳
  • Fourth Quadrant: Angles between 270掳 and 360掳


Each quadrant has a characteristic sign for the trigonometric functions: sine (sin), cosine (cos), and tangent (tan).
For instance, in the fourth quadrant, where angles range from 270掳 to 360掳, the signs are:

  • sin is negative
  • cos is positive
  • tan is negative


This information helps determine the signs of functions like cosecant (csc), secant (sec), and cotangent (cot) since they are reciprocals of sin, cos, and tan respectively.
angle measurement
Angle measurement is vital in trigonometry. Angles can be measured in degrees or radians.
When measuring in degrees, a full circle is 360掳. In radians, a full circle is \(2\pi\). It鈥檚 common to convert angles back and forth: divide degrees by 180 and multiply by \(\pi\) to get radians.

For instance, the angle 290掳 is a degree measurement.
To understand its position, examine the interval it falls into. Since 290掳 is over 270掳 but under 360掳, it is clear that it is in the fourth quadrant.

Angles are cyclical, meaning they wrap around every 360掳 in degrees or \(2\pi\) radians. So, 650掳 would be equivalent to (650 - 360 = 290掳), placing it back in the same fourth quadrant.
trigonometric function values
Trigonometric function values depend on angles and their quadrants. There are six primary trigonometric functions:

  • Sine (\text{sin})
  • Cosine (\text{cos})
  • Tangent (\text{tan})
  • Cosecant (\text{csc})
  • Secant (\text{sec})
  • Cotangent (\text{cot})


The values of these functions change based on the quadrant:
  • Sine (\text{sin}) represents the y-coordinate or vertical component \(\left( \frac{opposite}{hypotenuse} \right)\)
  • Cosine (\text{cos}) represents the x-coordinate or horizontal component \(\left( \frac{adjacent}{hypotenuse} \right)\)
  • Tangent (\text{tan}) is the ratio of \text{sin} to \text{cos} \(\left( \frac{opposite}{adjacent} \right)\)
  • Cosecant (\text{csc}) is the reciprocal of sine \(\left( \frac{hypotenuse}{opposite} \right)\)
  • Secant (\text{sec}) is the reciprocal of cosine \(\left( \frac{hypotenuse}{adjacent} \right)\)
  • Cotangent (\text{cot}) is the reciprocal of tangent \(\left( \frac{adjacent}{opposite} \right)\)

By understanding their interrelationships and quadrant signs, you can always determine their values accurately.

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Most popular questions from this chapter

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