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Chapter 4: Problem 70

In Exercises 69-74, find the indicated trigonometric value in the specified quadrant. Function \(cot\ \theta\ =\ -3\) Quadrant \(II\) Trigonometric Value \(sin\ \theta\)

Short Answer

Expert verified
\(\sqrt{\frac{1}{10}}\)

Step by step solution

01

Find the tangent value

Since \(cot\ \theta = -3\), then \(tan\ \theta = -\frac{1}{3}\) since tangent is the reciprocal of cotangent.
02

Find the value of cosine

From tangent, we know that \(tan\ \theta = \frac{sin\ \theta}{cos\ \theta}\), which implies \(cos\ \theta = \frac{sin\ \theta}{tan\ \theta}\). Since \(\theta\) is in the second quadrant where cosine is negative, we take the negative root and thus \(cos\ \theta = -\sqrt{1 - (sin\ \theta)^2}\).
03

Substitute cosine value

Substitute the value of cosine in the equation from step 2 to the equation \(tan\ \theta = \frac{sin\ \theta}{cos\ \theta}\), resulting in \(-\frac{1}{3} = \frac{sin\ \theta}{-\sqrt{1 - (sin\ \theta)^2}}\). Squaring both sides of the equation removes the square root and yields \(frac{1}{9} = \frac{(sin\ \theta)^2}{1 - (sin\ \theta)^2}\). Solve this for \(sin\ \theta\).
04

Solve for sine

Solving the equation gives \(sin\ \theta = \sqrt{\frac{1}{10}}\ or\ -\sqrt{\frac{1}{10}}\), but since sine is positive in the second quadrant, \(sin\ \theta = \sqrt{\frac{1}{10}}\)

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Key Concepts

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

Understanding Cotangent
Cotangent is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function. In mathematical terms, cotangent is defined as:
  • \[\cot \theta = \frac{1}{\tan \theta}\]
  • If the tangent of an angle \(\theta\) is \(-\frac{1}{3}\), then its cotangent, as given in our exercise, is \(-3\).
It is crucial to remember this relationship as it helps in solving many trigonometric problems. Cotangent can also be expressed in terms of sine and cosine:
  • \[\cot \theta = \frac{\cos \theta}{\sin \theta}\]
This expression shows that cotangent also intimately links the sine and cosine functions. Depending on which quadrant the angle is in, the signs of these trigonometric functions vary. Understanding these sign changes is key when working across different quadrants.
Exploring Sine
The sine of an angle is one of the most commonly encountered trigonometric functions. It is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
  • \[\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\]
However, in the context of the unit circle, it can also be seen as the y-coordinate of a point on the circle that corresponds to the angle \(\theta\).
  • Since the exercise specifies that \(\theta\) is in Quadrant II, the sine is positive.
  • This is because, in Quadrant II, the y-values (sine values) are positive while x-values (cosine values) are negative.
The sine function is essential in expressing a wide array of physical and geometric phenomena since it describes oscillations and waves so naturally.
Quadrant II Characteristics
When dealing with trigonometric functions, understanding the properties of quadrants is crucial. Quadrant II is one of the four sections of the unit circle, characterized by specific sign properties for trigonometric functions.
  • In this quadrant, sine and its reciprocal, cosecant, remain positive.
  • On the contrary, cosine, tangent, and their reciprocals (secant and cotangent) are negative.
This means whenever an angle \(\theta\) lies in Quadrant II, the corresponding value of sine is positive, which aligns with our given condition that \(\sin \theta = \sqrt{\frac{1}{10}}\).
  • Learning these sign conventions helps simplify the calculation of trigonometric values and avoid errors in determining the sign.
Remember, each quadrant has a unique set of positive and negative functions, so it鈥檚 a good practice to memorize these to bolster your trigonometric skills.

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