Problem 48 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

In a \(\triangle A B C\), if \(A=45^{\circ}, b=\sqrt{6}, a=2\), then find \(B\).

Step-by-Step Solution

Answer

In triangle ABC, angle B is approximately \(16.3^{\circ}\).

Step 1: Write down the given information

We are given: - In triangle ABC, - Angle A = 45° - Side b = √6 - Side a = 2 Our goal is to find angle B.

Step 2: Use the Law of Sines

The Law of Sines states that: \(\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}\) Since we know A, a, and b, we can use the formula to find angle B: \(\frac{\sin{A}}{a} = \frac{\sin{B}}{b}\) Plug in the given information: \(\frac{\sin{45}}{2} = \frac{\sin{B}}{\sqrt{6}}\)

Step 3: Solve for angle B

Now, we need to solve for angle B: \(\sin{B} = \frac{\sin{45} \cdot \sqrt{6}}{2}\) Calculate the value of sin(B): \(\sin{B} = \frac{\sqrt{6} / 2}{2} = \frac{\sqrt{6}}{4}\) Now, we can take the inverses sine (arcsine) of both sides to find angle B: \(B = \arcsin{\frac{\sqrt{6}}{4}}\) Use a calculator to find the value of angle B: \(B ≈ 16.3°\)

Step 4: Final Answer

In triangle ABC, angle B is approximately 16.3°.

Key Concepts

Law of Sines

The Law of Sines is a fundamental tool in trigonometry for solving triangles, particularly non-right triangles. This law relates the lengths of the sides of a triangle to the sines of its angles. The formula is expressed as:

• \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)

Here, \(A\), \(B\), and \(C\) are the angles of the triangle, while \(a\), \(b\), and \(c\) are the sides opposite these angles, respectively.

Using the Law of Sines allows us to find unknown angles and sides of a triangle when few measurements are already known. It's especially useful in oblique triangles where none of the angles is necessarily 90 degrees.

When applying this law to solve for an angle, such as angle \(B\), you substitute known values into the equation and solve for the sine of the unknown angle. Ensuring the correct ratio is maintained is crucial for accurate results.

Inverse Trigonometric Functions

Inverse trigonometric functions are the mathematical tools that allow us to find angles when given the value of a trigonometric function. When we calculate or solve \( \sin B \), we often need to reverse this process to find the actual angle \(B\).

The arcsine function, denoted as \(\arcsin\), is particularly helpful when dealing with sine values. If you know \( \sin{B} = \frac{\sqrt{6}}{4} \),

• To find the angle \(B\), you use the arcsine: \( B = \arcsin\left( \frac{\sqrt{6}}{4} \right) \).

This inverse function "undoes" the sine, returning us to the angle measurement itself.

It's important to note that for most calculations, especially involving a calculator, the ranges of inverse trigonometric functions are restricted to ensure they return a single, primary angle. This means the arcsine outputs an angle typically between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians, or equivalently between \(-90\degree\) and \(90\degree\) degrees in practical scenarios.

Angle Calculation

Calculating an angle in a triangle requires combining the use of various trigonometric principles, including the Law of Sines and inverse trigonometric functions, as detailed in previous sections.

In this exercise, to find angle \(B\) in triangle \(\triangle ABC\), we began with the Law of Sines and solved for \( \sin{B} \). After finding \( \sin{B} = \frac{\sqrt{6}}{4} \), our next step was to calculate the angle \(B\) using the arcsine function:

• \( B = \arcsin\left( \frac{\sqrt{6}}{4} \right) \)

This returned an approximate angle of \(16.3\degree\).

Remember, these calculations often rely on precise value inputs and understanding the function range limitations. Using a calculator can help ensure numerical accuracy, especially in converting between radians and degrees, when necessary.