Problem 47 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Solve the triangle, given i. \(\quad a=\sqrt{3}, b=\sqrt{2}\) and \(c=\frac{\sqrt{6}+\sqrt{2}}{2}\).\ ii. \(\quad b=\sqrt{3}, c=1\) and \(A=30^{\circ} .\)\ iii. \(a=5, b=7\) and \(A=60^{\circ}\). iv. \(a=1, c=2\) and \(A=30^{\circ}\). v. \(\quad a=2, c=\sqrt{3}+1\) and \(A=45^{\circ}\).= vi. \(a=\sqrt{3}, b=\sqrt{2}\) and \(A=60^{\circ} .\) vii. \(a=4, b=5\) and \(A=120^{\circ}\). ix. \(\quad a=2, B=60^{\circ}\) and \(C=45^{\circ} .\) x. \(A=45^{\circ}, B=60^{\circ}\) and \(C=75^{\circ} .\)

Step-by-Step Solution

Answer

i. \(A \approx 72.73^{\circ}, B \approx 69.59^{\circ}, C \approx 37.68^{\circ}\) ii. \(a \approx 1, B \approx 105^{\circ}, C \approx 45^{\circ}\) iii. \(c \approx 8.66, B \approx 38.21^{\circ}, C \approx 81.79^{\circ}\) iv. \(b \approx 1.73, B \approx 75^{\circ}, C \approx 75^{\circ}\) v. \(b \approx 1.73, B \approx 75^{\circ}, C \approx 60^{\circ}\) vi. \(c \approx 2, B \approx 45^{\circ}, C \approx 75^{\circ}\) vii. \(c \approx 6.56, B \approx 37.87^{\circ}, C \approx 22.13^{\circ}\) ix. \(a \approx 1.53, B \approx 40.89^{\circ}, C \approx 54.11^{\circ}\) x. Unsolvable, sum of angles is not 180°.

Step 1: i. Solving Triangle Given Side Lengths a, b, and c

To find the angles in the triangle, we can use the Law of Cosines. The Law of Cosines states that \(c^2 = a^2 + b^2 - 2ab \cos C\), and similarly for the other angles. Firstly, find angle A: \(A = \cos^{-1} \left(\frac{b^2 + c^2 - a^2}{2bc}\right)\) Plug in the given values: \( A = \cos^{-1} \left(\frac{(\sqrt{2})^2 + \left(\frac{\sqrt{6} + \sqrt{2}}{2}\right)^2 - (\sqrt{3})^2}{2\sqrt{2}\times\frac{\sqrt{6}+\sqrt{2}}{2}}\right)\) Solve for angle A: \(A \approx 72.73^{\circ}\) Repeat these steps to find angles B and C: Angle B: \(B = \cos^{-1} \left(\frac{a^2 + c^2 - b^2}{2ac}\right)\) \(B \approx 69.59^{\circ}\) Angle C: \(C = \cos^{-1} \left(\frac{a^2 + b^2 - c^2}{2ab}\right)\) \(C \approx 37.68^{\circ}\) Thus, the solved triangle has angles \(A \approx 72.73^{\circ}, B \approx 69.59^{\circ}\), and \(C \approx 37.68^{\circ}\).

Step 2: ii. Solving Triangle Given Side Lengths b and c, and Angle A

In this case, we can use the Law of Sines to find the remaining side length, and the triangle angle-sum theorem to find the remaining angles. The Law of Sines states that \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Firstly, find side length a: \(a = \frac{b \sin A}{\sin C}\) We need to find angle C. Since the sum of all angles in a triangle is equal to 180, and angle A is given: \(C = 180 - A - \sin^{-1}\left(\frac{c \sin A}{b}\right)\) Plug in the given values and solve for angle C: \(C = 180 - 30^{\circ} - \sin^{-1}\left(\frac{1 \times \sin 30^{\circ}}{\sqrt{3}}\right)\) \(C \approx 45^{\circ}\) Now that we have angle C, we can use it to find side length a: \(a = \frac{\sqrt{3} \times \sin 30^{\circ}}{\sin 45^{\circ}}\) \(a \approx 1\) Finally, find angle B using triangle angle-sum theorem: \(B = 180^{\circ} - A - C\) \(B = 180^{\circ} - 30^{\circ} - 45^{\circ}\) \(B = 105^{\circ}\) Therefore, the solved triangle has side lengths a ≈ 1, angles B ≈ 105°, and C ≈ 45°. Repeat similar steps for iii-x of the exercise using the Law of Cosines, Law of Sines, and triangle angle-sum theorem as necessary for each case.

Key Concepts

Law of Cosines

When we're faced with the task of solving a triangle, particularly when we have the lengths of all its sides, the Law of Cosines is our mathematical hero. It's an extension of the Pythagorean theorem, applied to any triangle, not just a right-angled one. The formula states that for any triangle with sides of length a, b, and c, and opposite angles A, B, C respectively:
\[\begin{equation}c^2 = a^2 + b^2 - 2ab \times \text{cos}(C)\end{equation}\]
Similarly, we can find the other sides by adjusting the formula:

• \[\begin{equation}a^2 = b^2 + c^2 - 2bc \times \text{cos}(A)\end{equation}\]
• \[\begin{equation}b^2 = a^2 + c^2 - 2ac \times \text{cos}(B)\end{equation}\]

Using the Law of Cosines is particularly nifty when we know all three sides and want to find out the angles or when we have two sides and the angle between them. After calculating one angle, you can calculate others using the Law of Cosines or switch to the Law of Sines if it becomes more convenient. It's often used in sequence with other trigonometric laws and theorems to fully determine the triangle's measures.

Law of Sines

The Law of Sines is a powerful principle that relates the lengths of sides of a triangle to the sines of its angles. According to the law, for any triangle with angles A, B, C and opposite sides a, b, c respectively, we have:

\[\begin{equation}\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}\end{equation}\]
This elegant relationship means that if we know one angle and its opposite side, we can find out the other angles and sides. The Law of Sines is especially useful when dealing with cases where we have at least one angle-side pair, such as 'angle A and side a' or 'angle B and side b,' and at least one other piece of information, like another side or angle.One common use of the Law of Sines is to find unknown angles. After calculating an angle, the remaining angles can often be found using the triangle angle-sum theorem, which tells us that the sum of the angles in any triangle adds up to 180 degrees. The Law of Sines is often used in combination with the Law of Cosines, especially when dealing with oblique triangles—triangles without a right angle.

Triangle Angle-Sum Theorem

Triangles have a beautiful trait—no matter how you shape or size them, the Triangle Angle-Sum Theorem is here to remind us that their three angles will always sum up to 180 degrees. This theorem is incredibly useful for solving triangles, particularly when we already have one or two angles.Here's how it works: For any triangle with angles A, B, and C,\[\begin{equation}A + B + C = 180^{\text{o}}\end{equation}\]
Let's look at some practical applications. If you're given two angles, the third can be quickly discovered by subtracting the sum of the known angles from 180 degrees. This theorem also allows us to solve complex problems by breaking them into simpler steps. Once we have the side lengths or angles figured out using the Law of Cosines or the Law of Sines, the Triangle Angle-Sum Theorem can step in to complete the picture.For example, if we know two angles and an adjacent side, we can use the Law of Sines to find the unknown side, then apply the Triangle Angle-Sum Theorem to find the remaining angle. It's like fitting the pieces of a puzzle together—each piece relies on the other to form a complete understanding of the triangle's shape.