Problem 71 - Precalculus with Limits

Question

In Exercises 67-72, (a) determine the quadrant in which \( u/2 \) lies, and (b) find the exact values of \( \sin(u/2) \), \( \cos(u/2) \), and \( \tan(u/2) \) using the half-angle formulas. \( \csc u = - \dfrac{5}{3}, \pi < u < \dfrac{3\pi}{2} \)

Step-by-Step Solution

Answer

The quadrant for \( u/2 \) is the second quadrant; \( \sin(u/2) = \sqrt{0.8} \), \( \cos(u/2) = \sqrt{0.1} \), \( \tan(u/2) = \sqrt{8} \).

Step 1: Determine the Quadrant

The Range of \( u \) is \( \pi < u < \dfrac{3\pi}{2} \), which puts \( u \) in the third quadrant. However, we are finding for \( u/2 \), which will be in the range \( \dfrac{\pi}{2} < u/2 < \dfrac{3\pi}{4} \). Thus, \( u/2 \) lies in the second quadrant.

Step 2: Find Sin \( u/2 \)

To get sin \( u/2 \), we use the half-angle formula where \( sin(x) = \sqrt{\dfrac{1 - cos(x)}{2}} \). However, we only have \( csc(u) \), so we will first find \( sin(u) = 1/csc(u) = -\dfrac{3}{5} \). Hence, \( sin(u/2) = \sqrt{\dfrac{1 - (-\dfrac{3}{5})}{2}} = \sqrt{\dfrac{4}{5}} = \sqrt{0.8} \). Since \( u/2 \) is in the second quadrant, the sine value is positive.

Step 3: Find Cos \( u/2 \)

Here, we use the half-angle formula for cosine \( cos(x) = \sqrt{\dfrac{1 + cos(x)}{2}} \). Applying the value of \( sin(u) \) we calculated before, we get \( cos(u) = \sqrt{1 - [sin(u)]^2} = \sqrt{1 - (-\dfrac{3}{5})^2} = -\dfrac{4}{5} \) as it's in the third quadrant. Hence, \( cos(u/2) = \sqrt{\dfrac{1 + (-\dfrac{4}{5})}{2}} = \sqrt{\dfrac{1}{10}} = \sqrt{0.1} \). Despite \( u/2 \) being in the second quadrant, where cosine is usually negative, we treat the angle as if it was in the first quadrant to get the principal square root.

Step 4: Find Tan \( u/2 \)

Tan is simply \( sin(x)/cos(x) \). Using the values obtained from the previous steps, we calculate \( tan(u/2) = sin(u/2)/cos(u/2) = \sqrt{0.8}/\sqrt{0.1} = \sqrt{8} \)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables included. They play a crucial role in simplifying complex trigonometric expressions, solving equations, and proving equivalencies. Among these identities, the half-angle formulas are particularly useful when dealing with angles that are half of a given angle.

The half-angle formulas are derived from the double-angle identities. They allow us to express trigonometric functions such as sine, cosine, and tangent in terms of half the angle:

• For sine: \( \sin\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 - \cos(u)}{2}} \)
• For cosine: \( \cos\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 + \cos(u)}{2}} \)
• For tangent: \( \tan\left(\frac{u}{2}\right) = \pm \sqrt{\frac{1 - \cos(u)}{1 + \cos(u)}} \)

These formulas simplify calculations when dealing with angles that are not standard angles found on the unit circle. When using these formulas, it's important to determine the quadrant in which the angle lies to apply the correct sign.

Understanding and mastering these identities is key to solving complex trigonometric problems efficiently.

Unit Circle

The unit circle is an essential concept in trigonometry that provides a geometric representation of the trigonometric functions. It's a circle with a radius of 1, centered at the origin of the coordinate plane. Understanding the unit circle helps us grasp how sine, cosine, and tangent relate to angles and is fundamental in studying trigonometric functions.

On the unit circle:

• The angle in radians corresponds to the length of the arc subtended by the radius.
• The coordinates of any point on the circle can be expressed as \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.

This setup allows us to visualize and derive the values for trigonometric functions easily. Moreover, each quadrant of the unit circle determines the sign of the trigonometric functions:

• In the first quadrant: both sine and cosine are positive.
• In the second quadrant: sine is positive, cosine is negative.
• In the third quadrant: both sine and cosine are negative.
• In the fourth quadrant: sine is negative, cosine is positive.

Using the unit circle in conjunction with trigonometric identities, students can better understand and remember the behavior of trigonometric functions across different angles.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are essential for understanding relationships involving angles and lengths in right triangles. These functions include sine, cosine, and tangent, each describing a specific ratio of sides in a right triangle. In the broader sense, these functions also describe periodic phenomena.

• Sine (\(\sin\)): It represents the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos\)): It indicates the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): It is the ratio of the opposite side to the adjacent side, or equivalently, the ratio of sine to cosine.

In the problem at hand, these trigonometric functions are evaluated at half-angles using specific formulas. The half-angle formulas are used when we know the value of one trigonometric function and need to find others for the half of the angle.

These functions can exhibit properties such as periodicity, symmetry, and specific values at standard angles, making them useful for modeling a vast array of real-world situations, including waves, oscillations, and circular motion.