Problem 67 - 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. \( \cos u = \dfrac{7}{25}, 0 < u < \dfrac{\pi}{2} \)

Step-by-Step Solution

Answer

\( u/2 \) lies in the first quadrant. \( \sin(u/2) = \dfrac{3\sqrt{2}}{5} \), \( \cos(u/2) = \dfrac{4\sqrt{2}}{5} \), \( \tan(u/2) = \dfrac{3}{4} \)

Step 1: Determine the quadrant of \( u/2 \)

From the given range of \( u \), \( 0 < u < \dfrac{\pi}{2} \), we note that \( u \) lies in the first quadrant. Therefore, \( u/2 \) will also lie in the first quadrant since dividing by 2 will decrease the value of \( u \) but will not make it negative or greater than \( \pi/2 \).

Step 2: Compute \( \sin(u/2) \)

We use the half-angle formula for sine: \( \sin(u/2) = \pm \sqrt{\dfrac{1-\cos u}{2}} \). Since \( u/2 \) is in the first quadrant where sine is always positive, we choose the positive square root. Substituting in the given value for \( \cos u = \dfrac{7}{25} \), we have \( \sin(u/2) = \sqrt{\dfrac{1-\dfrac{7}{25}}{2}} = \sqrt{\dfrac{18}{50}} = \dfrac{3\sqrt{2}}{5} \)

Step 3: Compute \( \cos(u/2) \)

We use the half-angle formula for cosine: \( \cos(u/2) = \pm \sqrt{\dfrac{1+\cos u}{2}} \). Since \( u/2 \) is in the first quadrant where cosine is always positive, we choose the positive square root. Substituting, we find \( \cos(u/2) = \sqrt{\dfrac{1+\dfrac{7}{25}}{2}} = \sqrt{\dfrac{32}{50}} = \dfrac{4\sqrt{2}}{5} \)

Step 4: Compute \( \tan(u/2) \)

The tangent of an angle is the ratio of the sine to the cosine of the angle. So, \( \tan(u/2) = \dfrac{\sin(u/2)}{\cos(u/2)} = \dfrac{3\sqrt{2}/5}{4\sqrt{2}/5} = \dfrac{3}{4} \)

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express relationships between the trigonometric functions. They are essential tools in solving trigonometric problems because they simplify expressions and allow us to understand connections between different trigonometric values. Two important identities are the Pythagorean Identity \( \sin^2 x + \cos^2 x = 1 \) and the Angle Sum Identities, which describe the sine and cosine of sums and differences of angles.
In this exercise, we focus on the half-angle identities. These are specific trigonometric identities used to find the sine, cosine, and tangent of half an angle, \( x/2 \), when the trigonometric functions of \( x \) are known. The half-angle formulas are as follows:

• \( \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos x}{2}} \)
• \( \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \)
• \( \tan\left(\frac{x}{2}\right) = \frac{\sin x}{1 + \cos x} \) or \( \frac{1 - \cos x}{\sin x} \)

These formulas are pivotal in calculating the exact values of trigonometric functions for half angles, especially when working with angles in specific quadrants.

Quadrants

The concept of quadrants is crucial in understanding the behavior of trigonometric functions. A quadrant refers to one of the four sections in a Cartesian coordinate system, divided by the x-axis and y-axis. Each quadrant has different characteristics concerning the signs of the trigonometric functions:

• **First Quadrant (0 to \( \frac{\pi}{2} \)):** All trigonometric functions are positive.
• **Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)):** Sine is positive, whereas cosine and tangent are negative.
• **Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)):** Tangent is positive, whereas sine and cosine are negative.
• **Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)):** Cosine is positive, whereas sine and tangent are negative.

In this exercise, since \( u \) is in the first quadrant, dividing by 2 keeps \( u/2 \) in the first quadrant as well. Knowing the quadrant helps determine the sign of the trigonometric functions, guiding the choice between positive or negative roots in the half-angle formulas.

Trigonometric Functions

Trigonometric functions relate the angles of triangles to the lengths of their sides and are fundamental in trigonometry. The main trigonometric functions include sine, cosine, and tangent, each having a specific relationship with the angle of a right triangle.

• **Sine (\( \sin \)):** Given an angle \( \theta \) in a right triangle, sine is the ratio of the length of the opposite side to the hypotenuse.
• **Cosine (\( \cos \)):** This function is the ratio of the length of the adjacent side to the hypotenuse.
• **Tangent (\( \tan \)):** Tangent is the ratio of the opposite side to the adjacent side, or equivalently, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In the context of half-angle identities, the exercise uses half-angle formulas to compute these functions for \( u/2 \). Having \( \cos u \) allows the use of half-angle identities to find \( \sin(u/2) \), \( \cos(u/2) \), and \( \tan(u/2) \), ensuring that the angle is understood within its correct quadrant.