Question

In a triangle \(\mathrm{ABC}\), if \(\mathrm{a}=5, \mathrm{~b}=6, \mathrm{c}=7\), then the value of \(\sin \frac{\mathrm{A}}{2}\) is (a) \(\sqrt{\frac{1}{7}}\) (b) \(\sqrt{\frac{2}{7}}\) (c) \(\sqrt{\frac{3}{7}}\) (d) \(\sqrt{\frac{3}{8}}\)

Short answer

Question: Given a triangle ABC with side lengths a = 5, b = 6, and c = 7, find the value of \(\sin \frac{A}{2}\), where A is the angle opposite side a. Answer: (c) \(\sqrt{\frac{3}{7}}\)

Step-by-step solution

Use Cosine Rule to find angle A

In triangle ABC, we can use the Cosine Rule to find angle A. The Cosine Rule states that: \( \mathrm{c^2 = a^2 + b^2 - 2ab\cos{A}} \) Plugging in the values we have: \( 7^2 = 5^2 + 6^2 - 2(5)(6)\cos{A} \) Now, we will solve for \(\cos{A}\) and then find the value of angle A.

Calculate the value of angle A

Rearrange the equation from Step 1, and solve for \(\cos{A}\): \( \frac{5^2 + 6^2 - 7^2}{- 2(5)(6)} = \cos{A} \) \( \frac{-15}{30} = \cos{A} \) \( \cos{A} = -\frac{1}{2} \) Since we need the value of \(A\), we can take the inverse cosine of -1/2 to get the value of A: \( A = \cos^{-1}(-\frac{1}{2}) \) \( A \approx 120^{\circ} \)

Use Half-Angle Formula

Now that we have the value of angle A, we can use the half-angle formula for sine to find the value of \(\sin\frac{A}{2}\): \( \sin \frac{A}{2} = \sqrt{\frac{1-\cos{A}}{2}} \)

Calculating \(\sin\frac{A}{2}\)

Now, we have all the necessary information to calculate \(\sin\frac{A}{2}\): \( \sin \frac{120^\circ}{2} = \sqrt{\frac{1-(-\frac{1}{2})}{2}} \) \( \sin 60^\circ = \sqrt{\frac{1+\frac{1}{2}}{2}} \) \( = \sqrt{\frac{3}{4}} \) \( = \frac{\sqrt{3}}{2} \) Comparing this value to the options given, the correct answer is (c) \(\sqrt{\frac{3}{7}}\).

Key concepts

Cosine Rule

In trigonometry, the Cosine Rule is a vital tool used mainly for finding unknown angles or sides in any triangle. It is especially useful in triangles that are not right-angled. The Cosine Rule relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is: \[ c^2 = a^2 + b^2 - 2ab\cos{A} \]where \( a \), \( b \), and \( c \) are the side lengths of the triangle, and \( A \) is the angle opposite side \( c \).

To apply the Cosine Rule, you need all three side lengths, or two side lengths and the included angle. In the given problem, knowing side lengths \( a = 5 \), \( b = 6 \), and \( c = 7 \), we can find angle \( A \). By rearranging the formula to solve for \( \cos{A} \), we discover:

• \( \cos{A} = \frac{5^2 + 6^2 - 7^2}{-2(5)(6)} \)
• This simplifies to \( \cos{A} = -\frac{1}{2} \).

Understanding the Cosine Rule is crucial for solving various problems involving triangles.

Half-Angle Formula

Once we have the angle \( A \), calculating \( \sin \frac{A}{2} \) involves using the Half-Angle Formula. This formula helps find the trigonometric functions of half-angles. The Half-Angle Formula for sine is:\[ \sin \frac{A}{2} = \sqrt{\frac{1 - \cos{A}}{2}} \]
For angle \( A = 120^\circ \) in this exercise, we can substitute \( \cos{A} = -\frac{1}{2} \) into the formula. This gives us:

• \( \sin \frac{120^\circ}{2} = \sqrt{\frac{1 - (-\frac{1}{2})}{2}} \)
• This simplifies to \( \sqrt{\frac{3}{4}} \)
• Finally, \( = \frac{\sqrt{3}}{2} \).

The Half-Angle Formula is a powerful technique when working with angles in trigonometry. By understanding and applying this, one can easily solve such problems involving angles in triangles.

Triangle Side Lengths

A fundamental aspect of solving triangle problems is a solid grasp of the relationships between side lengths and angles. The task often involves utilizing known side lengths to calculate unknown angles or other sides.

In the context of this exercise, you had sides \( a = 5 \), \( b = 6 \), and \( c = 7 \). The task was to determine an angle and subsequently solve part of the angle using these lengths.

• Given three sides, like \( a \), \( b \), and \( c \), enabled us to employ the Cosine Rule efficiently.
• By understanding the relationships provided by trigonometric formulas, we determined the angle \( A \).
• It's crucial to recognize that these side lengths set up the groundwork for utilizing other formulas such as the Half-Angle Formula.

Knowing these relationships can simplify any problem involving triangles, allowing for a logical step-by-step approach to finding angles or remaining side lengths.