Problem 15 - Algebra 2 and Trigonometry

Question

If \(\sin A=\frac{24}{25}\) and \(90^{\circ} < A < 180^{\circ}\), find: a. \(\sin \frac{1}{2} A\) b. \(\cos \frac{1}{2} A\) c. \(\tan \frac{1}{2} A\)

Step-by-Step Solution

Answer

\(\sin \frac{A}{2} = \frac{4}{5}, \cos \frac{A}{2} = \frac{3}{5}, \tan \frac{A}{2} = \frac{4}{3}\).

Step 1: Understanding the Problem

We need to find the half-angle trigonometric values for angle \(A\), where \(\sin A = \frac{24}{25}\) and \(90^{\circ} < A < 180^{\circ}\). We will use the half-angle formulas for sine, cosine, and tangent.

Step 2: Identifying the Quadrant

Since \(A\) is between \(90^{\circ}\) and \(180^{\circ}\), \(\sin A\) is positive and \(\cos A\) is negative.

Step 3: Calculate \(\cos A\)

Use the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\). We know \(\sin A = \frac{24}{25}\), so \(\sin^2 A = \frac{576}{625}\). Thus, \(\cos^2 A = 1 - \frac{576}{625} = \frac{49}{625}\). Since \(A\) is in the second quadrant, \(\cos A = -\frac{7}{25}\).

Step 4: Calculate \(\sin \frac{A}{2}\)

Use the formula \(\sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}}\), knowing \(\cos A = -\frac{7}{25}\), we find \(\sin \frac{A}{2} = \sqrt{\frac{1 + \frac{7}{25}}{2}} = \sqrt{\frac{32}{50}} = \sqrt{\frac{16}{25}} = \frac{4}{5}\).

Step 5: Calculate \(\cos \frac{A}{2}\)

Use the formula \(\cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}}\). Substitute \(\cos A = -\frac{7}{25}\), giving \(\cos \frac{A}{2} = \sqrt{\frac{1 - \frac{7}{25}}{2}} = \sqrt{\frac{18}{50}} = \sqrt{\frac{9}{25}} = \frac{3}{5}\).

Step 6: Calculate \(\tan \frac{A}{2}\)

Use the formula \(\tan \frac{A}{2} = \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}}\). With \(\sin \frac{A}{2} = \frac{4}{5}\) and \(\cos \frac{A}{2} = \frac{3}{5}\), we have \(\tan \frac{A}{2} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}\).

Key Concepts

Half-angle formulas

Half-angle formulas are a set of trigonometric identities useful for finding the sine, cosine, and tangent of half of a given angle. These formulas are especially handy when the original angle is in a standard trigonometric form. They allow us to calculate the values without resorting to complex calculations.

• For sine, the formula is: \(\sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}}\).
• For cosine, use: \(\cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}}\).
• And for tangent: \(\tan \frac{A}{2} = \frac{\sin A}{1 + \cos A}\) or \(\tan \frac{A}{2} = \frac{1 - \cos A}{\sin A}\).

These equations derive from the double-angle identities, allowing us to tackle problems involving angles not directly found in standard trigonometric tables. By simplifying and applying these formulas, we can efficiently determine the trigonometric ratios for half of any angle.

Pythagorean identity

The Pythagorean identity is one of the foundational cornerstones in trigonometry. It expresses the intrinsic relationship between the sine and cosine of an angle:
\[\sin^2 A + \cos^2 A = 1\]
When given one trigonometric function, we can determine the other using this identity. In our problem, we know \(\sin A = \frac{24}{25}\), which helps us find \(\cos A\) by rearranging the identity:
\[\cos^2 A = 1 - \sin^2 A\]
Calculating, we find \(\cos A\) to be \(-\frac{7}{25}\) since \(A\) is in the second quadrant, where cosine values are negative. The Pythagorean identity proves indispensable in a variety of trigonometric problems and remains a vital tool for any student mastering trigonometry.

Trigonometric ratios

Trigonometric ratios involve the relationships between the angles and sides of a triangle, particularly in right-angled triangles. These ratios are essential for understanding the properties of waves, circles, and many other geometrical features.
The primary trigonometric ratios are sine, cosine, and tangent, which represent the following:

• Sine (\(\sin\)): Ratio of opposite side to the hypotenuse.
• Cosine (\(\cos\)): Ratio of adjacent side to the hypotenuse.
• Tangent (\(\tan\)): Ratio of opposite side to adjacent side.

In the context of solving our problem, after determining \(\sin \frac{A}{2}\) and \(\cos \frac{A}{2}\), the tangent of half the angle becomes simply the ratio of these two, computed as \(\tan \frac{A}{2} = \frac{4}{3}\). Recognizing these relationships is essential for calculating and simplifying more complex expressions involving angles.

Second quadrant

Understanding the quadrants in which angles reside is essential to correctly determining the sign of trigonometric functions. The second quadrant ranges from \(90^{\circ}\) to \(180^{\circ}\), and each trigonometric function has distinct properties within that range.

• In the second quadrant, the sine of an angle is positive.
• The cosine and tangent, however, are negative.

In our specific exercise, knowing that the angle \(A\) lies in this quadrant directly affects our calculations and conclusions. For example, while calculating \(\cos A\), we determine that it should be negative, yielding \(-\frac{7}{25}\). Additionally, understanding these quadrant rules helps to accurately apply half-angle formulas, ensuring we account for the correct positive or negative value of each function after halving the angle. This knowledge is crucial for avoiding common mistakes in trigonometry.