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Chapter 5: Problem 17

In Exercises 17-26, use the half-angle identities to find the desired function values. $$ \text { If } \cos x=\frac{5}{13} \text { and } \frac{3 \pi}{2}

Short Answer

Expert verified
\( \sin\left(\frac{x}{2}\right) = \frac{2\sqrt{13}}{13} \) is positive, as \( x/2 \) is in the second quadrant.

Step by step solution

01

Identify the Half-Angle Identity for Sine

The half-angle identity for sine is given by:\[ \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}} \]We need to determine the sign, but determining this will be dealt with in future steps.
02

Calculate the Expression Inside the Square Root

Substitute the given value of \( \cos x = \frac{5}{13} \) into the half-angle identity:\[ \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \frac{5}{13}}{2}} \]Simplify the expression inside the square root:\[ \frac{1 - \frac{5}{13}}{2} = \frac{\frac{13}{13} - \frac{5}{13}}{2} = \frac{\frac{8}{13}}{2} = \frac{4}{13} \]
03

Evaluate the Square Root

Now we take the square root of the expression found in the previous step:\[ \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{4}{13}} = \pm \frac{2}{\sqrt{13}} \]To rationalize the denominator, multiply numerator and denominator by \( \sqrt{13} \):\[ \sin\left(\frac{x}{2}\right) = \pm \frac{2\sqrt{13}}{13} \]
04

Determine the Correct Sign for the Result

Since \( \frac{3\pi}{2} < x < 2\pi \), \( x \) is in the fourth quadrant where cosine is positive and sine is negative. When considering \( \frac{x}{2} \), it falls in the second quadrant, making sine positive in this context.Thus, \( \sin\left(\frac{x}{2}\right) \) should be positive:\[ \sin\left(\frac{x}{2}\right) = \frac{2\sqrt{13}}{13} \]

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are essential tools in understanding the relationships between different trigonometric functions. These are equations that hold true for every value of the variable where both sides of the equation are defined. In this exercise, we specifically use the half-angle identity for sine to find the function value.
  • Half-angle identities allow us to calculate the sine, cosine, or tangent of half an angle from the full angle.
  • They are useful in simplifying expressions and solving trigonometry problems involving angles.
For instance, the identity for sine we used is:\[ \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}} \]This identity can present two possible signs, which depend on the resulting quadrant for the angle \(\frac{x}{2}\). Understanding these identities helps students manipulate and transform trigonometric expressions efficiently.
Sine Function
The sine function is one of the primary trigonometric functions and is fundamental in describing wave behaviors, oscillations, and circular motion. It varies in value from -1 to 1 as the angle changes. In this exercise, we are interested in finding the sine of half an angle.
  • The sine function of an angle is the y-coordinate of a point on the unit circle at that angle.
  • For acute angles, sine can be perceived as the opposite side of a right triangle divided by the hypotenuse.
It is crucial to note how the sine function behaves differently depending on the angle's location in the trigonometric circle's quadrants. Quadrant locations affect whether the sine function outputs a positive or negative value.
Cosine Function
The cosine function complements the sine function in trigonometry and also ranges from -1 to 1. Cosine values correspond to the x-coordinate on the unit circle, giving insight into horizontal projections of angles.
  • In practice, cosine is the adjacent side of a right triangle divided by the hypotenuse.
  • For the angle \(x\) provided in the problem where \(\cos x = \frac{5}{13}\), this indicates a relatively small negative sine value at \(x\), typical for angles in certain quadrants like the fourth.
In this particular problem, cosine is used to determine sine explicitly using the half-angle identity. The two functions are interdependent in many trigonometric identities and problems.
Quadrants in Trigonometry
Understanding the quadrants in trigonometry is key to comprehending how angles and function values change around the unit circle. The unit circle is divided into four quadrants, each affecting the sign of sine, cosine, and tangent values.
  • Quadrant I: Both sine and cosine are positive.
  • Quadrant II: Sine is positive, cosine is negative.
  • Quadrant III: Both sine and cosine are negative.
  • Quadrant IV: Sine is negative, cosine is positive.
This exercise provides \(x\) in the fourth quadrant where cosine values are positive and sine values are negative. However, since we're finding \(\sin\left(\frac{x}{2}\right)\), and understanding its location in the second quadrant, sine is positive there, demonstrating how quadrant knowledge is crucial for selecting the correct sign in trigonometric identities.

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Most popular questions from this chapter

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