Problem 45 - Precalculus with Calculus Previews

Question

Use the given information to find (a) \(\cos (x / 2)\), (b) \(\sin (x / 2)\), and (c) \(\tan (x / 2)\). $$ \sin x=\frac{12}{13}, \pi / 2

Step-by-Step Solution

Answer

(a) \(\cos(x/2) = \frac{2}{\sqrt{13}}\), (b) \(\sin(x/2) = \frac{3}{\sqrt{13}}\), (c) \(\tan(x/2) = \frac{3}{2}\).

Step 1: Understand the Trigonometric Context

We are given \( \sin x = \frac{12}{13} \), which suggests a right triangle where the opposite side to angle \( x \) is 12 and the hypotenuse is 13. Since \( \pi/2 < x < \pi \) (second quadrant), \( \cos x \) will be negative.

Step 2: Find \( \cos x \)

Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), we solve for \( \cos x \):\[ \cos^2 x = 1 - \sin^2 x = 1 - \left(\frac{12}{13}\right)^2 = \frac{25}{169} \]Thus, \( \cos x = -\frac{5}{13} \) since \( x \) is in the second quadrant where cosine is negative.

Step 3: Calculate \( \cos(x/2) \) using Half-Angle Formulas

The half-angle formula for cosine is \( \cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos x}{2}} \).Since \( x \) is in the second quadrant, \( \frac{x}{2} \) is in the first quadrant, so we take the positive square root.\[ \cos \left( \frac{x}{2} \right) = \sqrt{\frac{1 - 5/13}{2}} = \sqrt{\frac{8/13}{2}} = \sqrt{\frac{4}{13}} = \frac{2}{\sqrt{13}} \]

Step 4: Calculate \( \sin(x/2) \) using Half-Angle Formulas

The half-angle formula for sine is \( \sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}} \).Again, since \( \frac{x}{2} \) is in the first quadrant, we take the positive square root.\[ \sin \left( \frac{x}{2} \right) = \sqrt{\frac{1 + 5/13}{2}} = \sqrt{\frac{18/13}{2}} = \sqrt{\frac{9}{13}} = \frac{3}{\sqrt{13}} \]

Step 5: Calculate \( \tan(x/2) \) using the Sine and Cosine Results

The tangent can be found using \( \tan \left( \frac{x}{2} \right) = \frac{\sin(x/2)}{\cos(x/2)} \).\[ \tan \left( \frac{x}{2} \right) = \frac{\frac{3}{\sqrt{13}}}{\frac{2}{\sqrt{13}}} = \frac{3}{2} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools for solving various problems in trigonometry. They provide relationships between the basic trigonometric functions such as sine (\(\sin(x)\)), cosine (\(\cos(x)\)), and tangent (\(\tan(x)\)).These identities allow us to simplify complex expressions and solve equations. Some of the most utilized trigonometric identities include:

• Reciprocal identities
• Pythagorean identities
• Angle sum and difference identities
• Half-angle identities
• Double angle identities

For example, using the Pythagorean identity \(\sin^2x + \cos^2x = 1\), you can find one trigonometric value if you are given the other, making it very useful in various exercises, such as the one provided.In the context provided, trigonometric identities such as the half-angle formula are crucial. They allow us to find the values of trigonometric functions at half of a given angle. This is essential because knowing these formulas ensures we're equipped to tackle a wide range of trigonometric problems efficiently.

Right Triangle

The concept of a right triangle is central to understanding trigonometric functions. In a right triangle, one angle is always 90 degrees. The other two angles will always add up to 90 degrees, making them complementary.The sides of a right triangle are:

• The hypotenuse: the side opposite the right angle and the longest side
• The opposite side: the side opposite the angle you are referencing
• The adjacent side: the side next to the angle you are referencing but not the hypotenuse

The trigonometric functions are ratios of these sides:

• \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

In this exercise, knowing that \(\sin x = \frac{12}{13}\) hints at a right triangle where 12 is the opposite side and 13 is the hypotenuse. This helps in constructing the triangle and finding the other sides using the Pythagorean Theorem.

Second Quadrant

The unit circle, which is fundamental to trigonometry, is divided into four quadrants. The placement of an angle within these quadrants impacts the sign of its trigonometric functions. When an angle is between \(90^{\circ}\) and \(180^{\circ}\), it is in the second quadrant.Key characteristics of angles in the second quadrant are:

• \(\sin(x)\) is positive
• \(\cos(x)\) is negative
• \(\tan(x)\) is negative

This explains why in the problem given, where \(\pi/2 < x < \pi\), the cosine of \(x\) turns out to be negative. Recognizing to which quadrant an angle belongs helps in determining the signs of sine, cosine, and tangent, which is critical in calculations like utilizing the half-angle formulas.

Pythagorean Identity

The Pythagorean identity is one of the most crucial trigonometric identities, derived from the Pythagorean theorem. It states that for any angle \(x\) on the unit circle, the sum of the squares of sine and cosine equals 1: \[\sin^2(x) + \cos^2(x) = 1\]This identity is built upon the idea that a point's distance from the origin on the unit circle is always 1. Since the hypotenuse (the radius of the unit circle) is 1, this equation naturally holds true.Using this identity, if you know either the sine or cosine of an angle, you can easily solve for the other. In the exercise, \(\sin x = \frac{12}{13}\) is used in the Pythagorean identity to find \(\cos x\):\[\cos^2 x = 1 - \left(\frac{12}{13}\right)^2\]This results in \(\cos^2 x = \frac{25}{169}\), and further,\(\cos x = -\frac{5}{13}\) is derived considering the quadrant. Remembering these identities is essential for solving trigonometric equations and simplifying complex expressions.