Problem 70 - 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. \( \cot u = 3, \pi < u < \dfrac{3\pi}{2} \)

Step-by-Step Solution

Answer

\( u/2 \) resides in the second quadrant, and the exact values for \( \sin(u/2) \), \( \cos(u/2) \), and \( \tan(u/2) \) are \( \sqrt{2 - \sqrt{2}}/2 \), \( -\sqrt{2 + \sqrt{2}}/2 \), and \( \sqrt{2 - \sqrt{2}} \), respectively.

Step 1: Identify Quadrant for u/2

Given \( \pi < u < \dfrac{3\pi}{2} \), which lies in the third quadrant. To determine the quadrant in which \( u/2 \) lies, divide these intervals by 2, which leads to \( \dfrac{\pi}{2} < u/2 < \dfrac{3\pi}{4} \). This is in the second quadrant, where sine is positive and cosine is negative.

Step 2: Calculate cos u

Given \( \cot u = 3 \), and that cotangent is the reciprocal of tangent, we can write \( \tan u = \dfrac{1}{3} \). The identity \( \cos^2 u + \sin^2 u = 1 \) can be re-arranged to \( \cos u = \sqrt{1 - \sin^2 u} \). In the third quadrant, cosine is negative, so \( \cos u = -\sqrt{1 - \tan^2 u} = -\sqrt{1 - (1/3)^2} = -\sqrt{8/9} = -2\sqrt{2}/3 \).

Step 3: Calculate sin(u/2), cos(u/2), tan(u/2)

Utilise the half-angle formulas, which are: \( \sin(u/2) = \pm \sqrt{\dfrac{1 - \cos u}{2}} \), \( \cos(u/2) = \pm \sqrt{\dfrac{1 + \cos u}{2}} \), \( \tan(u/2) = \pm \sqrt{\dfrac{1 - \cos u}{1 + \cos u}} \). Note the signs depend on the quadrant where the angle resides. Based on the previous steps, we already know \( u/2 \) is in the second quadrant where sine is positive and cosine is negative. Thus, calculate the results: \( \sin(u/2) = \sqrt{\dfrac{1 - (-2\sqrt{2}/3)}{2}} = \sqrt{2 - \sqrt{2}}/2 \), \( \cos(u/2) = -\sqrt{\dfrac{1 + (-2\sqrt{2}/3)}{2}} = -\sqrt{2 + \sqrt{2}}/2 \), \( \tan(u/2) = \sqrt{\dfrac{1 - (-2\sqrt{2}/3)}{1 + (-2\sqrt{2}/3)}} = \sqrt{2 - \sqrt{2}} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in geometry, helping to simplify and solve expressions involving trigonometric functions. These identities reveal various relationships between sine, cosine, tangent, cotangent, secant, and cosecant. Among the numerous trigonometric identities, the Pythagorean identities take a central place, with the fundamental one expressed as:
\( \cos^2 u + \sin^2 u = 1 \).
This identity is highly useful, especially when paired with reciprocal identities such as cotangent. For instance, if \( \cot u = 3 \), we can deduce \( \tan u = \frac{1}{3} \), allowing us to build upon these relationships to find other trigonometric values.

• Reciprocal identities relate quotients of functions, enhancing our calculations, such as \( \tan u = \frac{1}{\cot u} \).
• Half-angle identities are derivatives of these core identities and help find function values at half the existing angle. For example, they are denoted as \( \sin(u/2) = \pm \sqrt{\frac{1 - \cos u}{2}} \).

Grasping these identities is crucial as they simplify trigonometric expressions, making complex calculations much more approachable.

Quadrants of the Unit Circle

The unit circle is divided into four quadrants, aiding in determining the sign of trigonometric functions. Each quadrant represents a specific range of angles:

• First Quadrant: \(0 < \theta < \frac{\pi}{2}\)
• Second Quadrant: \(\frac{\pi}{2} < \theta < \pi\)
• Third Quadrant: \(\pi < \theta < \frac{3\pi}{2}\)
• Fourth Quadrant: \(\frac{3\pi}{2} < \theta < 2\pi\)

The quadrant determines whether sine, cosine, and tangent values are positive or negative. In the second quadrant, sine remains positive, whereas cosine and tangent are negative. Knowing \( u \) lies between \( \pi \) and \( \frac{3\pi}{2} \), which places it in the third quadrant.
For the problem provided, dividing this range by 2 gives \( u/2 \) as lying in \( \frac{\pi}{2} < u/2 < \frac{3\pi}{4} \), positioning it in the second quadrant where sine is positive, and cosine is negative. This quadrant analysis is vital for applying trigonometric calculations, especially when using half-angle formulas.

Cotangent

Cotangent (\( \cot \)) is one of the fundamental trigonometric functions, defined as the reciprocal of tangent:
\( \cot u = \frac{1}{\tan u} = \frac{\cos u}{\sin u} \).
This function is particularly useful when dealing with angles and their relationships on the unit circle and can simplify calculations significantly when combined with other trigonometric identities. In the given problem, knowing \( \cot u = 3 \) helps derive \( \tan u = \frac{1}{3} \).

• Cotangent is positive where tangent is positive due to its reciprocal nature.
• It’s negative where tangent is negative, influencing the calculations for related angles.

Understanding cotangent and its properties allows students to navigate between different trigonometric identities more fluidly, unlocking solutions even in complex scenarios. For the exercise spanning parts (a) and (b), leveraging cotangent provided the necessary clue to unravel the hemisphere this angle falls in, guaranteeing correct application of the sought values using the half-angle formulas.