Problem 52 - Algebra and Trigonometry

Question

\(51-54\). Evaluate each expression under the given conditions. $$ \sin (\theta / 2) ; \tan \theta=-\frac{5}{12}, \theta \text { in Quadrant IV } $$

Step-by-Step Solution

Answer

\(\sin(\theta/2) = \frac{\sqrt{26}}{26}\).

Step 1: Identify the Triangle Properties

Since \(\tan \theta = -\frac{5}{12}\) and \(\theta\) is in Quadrant IV, where tangent is negative, this tells us the opposite side is 5 and the adjacent side is 12, with a hypotenuse calculated as 13 using the Pythagorean theorem: \(5^2 + 12^2 = 25 + 144 = 169 = 13^2\).

Step 2: Verify the Quadrant and Sign of Cosine

For angles in Quadrant IV, cosine is positive. Using this, calculate cosine: \(\cos \theta = \frac{adjacent}{hypotenuse} = \frac{12}{13}\).

Step 3: Apply the Half-Angle Formula

The half-angle formula for sine is \( \sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}} \). Since \(\theta/2\) in Quadrant IV is actually in Quadrant II (where sine is positive), use the positive square root.

Step 4: Compute the Value of Sine Half-Angle

Substitute the value of \(\cos \theta\) into the formula: \( \sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{12}{13}}{2}} = \sqrt{\frac{1/13}{2}} = \sqrt{\frac{1}{26}} = \frac{1}{\sqrt{26}} \).

Step 5: Rationalize the Denominator

To get the final result in standard form, rationalize the denominator: \(\frac{1}{\sqrt{26}} \cdot \frac{\sqrt{26}}{\sqrt{26}} = \frac{\sqrt{26}}{26}\).

Key Concepts

Half-Angle Formulas

Half-angle formulas are essential in trigonometry, providing us a way to express the trigonometric functions of half angles. They are useful for simplifying expressions and solving equations involving half angles. The half-angle formulas allow us to find the sine, cosine, and tangent of half an angle based on the known values of sine, cosine, or tangent of the initial angle. Here's an example for sine:\[\sin \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}\]In practice, this formula allows us to determine the value of \( \sin(\theta/2) \) when \( \cos \theta \) is known. The sign (positive or negative) depends on the quadrant where \( \theta/2 \) is located. Let's break it down:

• Start with a known value of \( \theta \).
• Determine \( \cos \theta \).
• Substitute into the formula.
• Adjust the sign based on the quadrant of \( \theta/2 \).

In our exercise, since \( \theta/2 \) is located in Quadrant II, where sine is positive, we opt for the positive square root. Utilizing these formulas helps us tackle complex trigonometric problems by reducing them to familiar forms.

Right Triangle Trigonometry

Right triangle trigonometry is the foundation of understanding trigonometric functions and their relationships. In a right triangle, the sides and angles have a specific relationship described by trigonometric ratios: sine, cosine, and tangent. The basic formulas are:

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

In the given problem, the tangent of angle \( \theta \) is \(-\frac{5}{12}\). This indicates that in our right triangle:

• The side opposite \( \theta \) is \( 5 \).
• The side adjacent to \( \theta \) is \( 12 \).

By applying the Pythagorean theorem, \( a^2 + b^2 = c^2 \), with \( a = 5 \) and \( b = 12 \), the hypotenuse (\( c \)) is found to be \( 13 \). This relationship is vital as it allows us to compute any trigonometric function, given sufficient initial conditions, aiding in solving more complex problems.

Coordinate System Quadrants

The coordinate system is divided into four quadrants, each with unique properties affecting the signs of trigonometric functions. Understanding which quadrant an angle lies in helps determine whether these functions are positive or negative.The quadrants are defined as follows:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, cosine and tangent are negative.
• Quadrant III: Tangent is positive, sine and cosine are negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

In our scenario, \( \theta \) is located in Quadrant IV. Here, \( \tan \theta \) is negative, which aligns with the property that only cosine is positive in this quadrant. However, when using the half-angle formulas, it is crucial to determine which quadrant \( \theta/2 \) falls into, as it influences the sign of our result.For instance, if \( \theta \) is in Quadrant IV, \( \theta/2 \) might fall into Quadrant II, where only sine is positive. This understanding is critical for accurately applying trigonometric identities and formulas.