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Chapter 8: Problem 10

Determine the sign of the given functions. $$\sin 335^{\circ}, \cot 265^{\circ}$$

Short Answer

Expert verified
\( \sin 335^{\circ} \) is negative; \( \cot 265^{\circ} \) is positive.

Step by step solution

01

Identify the Quadrant

First, determine the quadrant in which each angle lies. The angle for \( \sin 335^{\circ} \) is measured from the positive x-axis and moves counterclockwise. Since \( 335^{\circ} \) is more than \( 270^{\circ} \) but less than \( 360^{\circ} \), it falls in the fourth quadrant.Similarly, for \( \cot 265^{\circ} \), since \( 265^{\circ} \) is more than \( 180^{\circ} \) but less than \( 270^{\circ} \), it falls in the third quadrant.
02

Apply Sign Rules for Quadrants

Each quadrant has specific sign rules for trigonometric functions:- In the fourth quadrant, \( \sin \) is negative.- In the third quadrant, \( \cot \) is positive.Based on these rules, \( \sin 335^{\circ} \) is negative, and \( \cot 265^{\circ} \) is positive.

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Key Concepts

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

Quadrantal Angles
Quadrantal angles are angles that are exactly at the boundaries of the four quadrants in a standard coordinate plane. These angles are key points usually represented by 0掳, 90掳, 180掳, 270掳, and 360掳.
When dealing with non-quadrantal angles such as 335掳 or 265掳, it's important to determine which quadrant they fall into.
  • Quadrant I ranges from 0掳 to 90掳
  • Quadrant II from 90掳 to 180掳
  • Quadrant III from 180掳 to 270掳
  • Quadrant IV from 270掳 to 360掳
Understanding which quadrant an angle lies in is crucial because the sign of trigonometric functions change depending on the quadrant. For example, the angle 335掳 falls in the fourth quadrant, while 265掳 is in the third quadrant. Knowing this helps determine the sign of the trigonometric functions involved.
Sine Function
The sine function is one of the fundamental trigonometric functions. It expresses the vertical component of an angle in the unit circle.

Definition

The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the unit circle, it corresponds to the y-coordinate of the point where the angle鈥檚 terminal side intersects the circle.

Signs in Different Quadrants

Understanding the sine function's behavior across various quadrants is crucial:
  • In Quadrant I, sine is positive because both x and y are positive, and the hypotenuse is always positive.
  • In Quadrant II, sine remains positive since y is still positive.
  • In Quadrant III, sine becomes negative due to the negative y-values.
  • Finally, in Quadrant IV, as is the case with \sin 335^\circ, sine is negative once more.
When calculating \( \sin 335^\circ \), its position in the fourth quadrant directly informs us that it will be negative.
Cotangent Function
Cotangent, another trigonometric function, is the reciprocal of tangent. It can be understood through its relationship with the tangent function and in reference to the coordinate plane.

Definition and Calculation

The cotangent of an angle can be defined as the ratio of the adjacent side to the opposite side in a right triangle. Alternatively, in terms of tangent, it is the reciprocal: \[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\]

Sign Behavior by Quadrant

The sign variation of cotangent across quadrants affects calculations:
  • Quadrant I: both sine and cosine are positive, thus cotangent is positive.
  • Quadrant II: cotangent becomes negative because cosine is negative while sine is positive.
  • Quadrant III: both sine and cosine are negative, making cotangent positive.
  • Quadrant IV: cosine is positive and sine is negative, resulting in a negative cotangent.
For \(\cot 265^\circ\), positioned in the third quadrant, we find cotangent positive as both sine and cosine are negative.

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Most popular questions from this chapter

Solve the given problems. In Exercises 53 and 54 assume \(\left.0^{\circ}<\theta<90^{\circ} . \text { (Hint: Review cofunctions on page } 125 .\right)\) Using the fact that cot \(20^{\circ}=2.747,\) evaluate tan \(290^{\circ}\)

solve the given problems. An ammeter needle is deflected \(52.00^{\circ}\) by a current of \(0.2500 \mathrm{A}\) The needle is 3.750 in. long, and a circular scale is used. How long is the scale for a maximum current of \(1.500 \mathrm{A} ?\)

Solve the given problems. In Exercises 53 and 54 assume \(\left.0^{\circ}<\theta<90^{\circ} . \text { (Hint: Review cofunctions on page } 125 .\right)\) $$\text { Using the fact that } \sin 75^{\circ}=0.9659, \text { evaluate } \cos 195^{\circ}$$

Find the trigonometric functions of \(\theta\) if the terminal side of \(\theta\) passes through the given point. All coordinates are exact. $$(-5,5)$$

Another use of radians is illustrated. Use a calculator (in radian mode) to evaluate the ratios \((\sin \theta) / \theta\) and \((\tan \theta) / \theta\) for \(\theta=0.1,0.01,0.001,\) and \(0.0001 .\) From these values, explain why it is possible to say that $$\sin \theta=\tan \theta=\theta$$ approximately for very small angles measured in radians.
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