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Chapter 6: Problem 76

Find the signs of the six trigonometric function values for the given angles. $$-620^{\circ}$$

Short Answer

Expert verified
In the second quadrant: sin and csc are positive; cos, sec, tan, cot are negative

Step by step solution

01

Convert the Angle to a Positive Coterminal Angle

To find the coterminal angle of \( -620^{\circ} \), add 360掳 repeatedly until the angle is between 0掳 and 360掳:\( -620^{\circ} + 2 \times 360^{\circ} = -620^{\circ} + 720^{\circ} = 100^{\circ} \). So, the positive coterminal angle is 100掳.
02

Determine the Quadrant

The coterminal angle \(100^{\circ} \) lies in the second quadrant because it is between 90掳 and 180掳.
03

Determine the Signs of the Trigonometric Functions in that Quadrant

In the second quadrant: - Sine (sin) is positive - Cosine (cos) is negative - Tangent (tan) is negative - Cosecant (csc) is positive (since it's the reciprocal of sine) - Secant (sec) is negative (since it's the reciprocal of cosine) - Cotangent (cot) is negative (since it's 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 Function Signs
Understanding the signs of trigonometric functions in various quadrants is crucial for solving trigonometry problems. Each quadrant in the coordinate plane affects the sine, cosine, tangent, cosecant, secant, and cotangent values differently.

Here鈥檚 a quick guide:
  • First Quadrant (0掳 to 90掳): All six trigonometric functions are positive.
  • Second Quadrant (90掳 to 180掳): Sine and cosecant are positive, while cosine, secant, tangent, and cotangent are negative.
  • Third Quadrant (180掳 to 270掳): Tangent and cotangent are positive, while sine, cosecant, cosine, and secant are negative.
  • Fourth Quadrant (270掳 to 360掳): Cosine and secant are positive, while sine, cosecant, tangent, and cotangent are negative.

For the angle of 100掳, which lies in the second quadrant, the sine and cosecant functions will be positive, while the others will be negative. Knowing these quadrant rules helps in quickly identifying the sign of any trigonometric function.
Angle Conversion
Converting angles into their coterminal angles is a fundamental skill in trigonometry. To find a positive coterminal angle, add or subtract multiples of 360掳 until you have an angle between 0掳 and 360掳.

Let鈥檚 see an example:
  • Starting angle: -620掳
  • Add 360掳 twice: \(-620^\text{掳} + 2 \times 360^\text{掳} = -620^\text{掳} + 720^\text{掳} = 100^\text{掳}\)

Now we have a positive coterminal angle of 100掳.

Converting angles like this makes it easier to determine their position in the coordinate plane.
Quadrant Determination
Knowing which quadrant an angle lies in is integral to understanding trigonometric functions. 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
For example, an angle of 100掳 falls between 90掳 and 180掳, placing it in the second quadrant.

Once you know the quadrant, you can determine the signs of the trigonometric functions. This becomes straightforward when paired with the understanding of how each function behaves in different quadrants.

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Most popular questions from this chapter

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