Question

\text { In a } \triangle \mathrm{ABC} \text { , if } \mathrm{a}^{4}-2\left(\mathrm{~b}^{2}+\mathrm{c}^{2}\right) \mathrm{a}^{2}+\mathrm{b}^{4}+\mathrm{b}^{2} \mathrm{c}^{2}+\mathrm{c}^{4}=0 \text { prove that } \tan \mathrm{A}=\pm \sqrt{3}

Short answer

Question: Prove that for any triangle ABC with side lengths a, b, and c, if a^4 - 2(b^2 + c^2)a^2 + b^4 + b^2c^2 + c^4 = 0, then tan(A) = ±√3.

Step-by-step solution

Rearrange the given equation

We can rearrange the given equation: a^4 - 2(b^2 + c^2)a^2 + b^4 + b^2c^2 + c^4 = 0 as: (a^2 - b^2 - c^2)^2 = b^2c^2(b^2 + c^2)

Relate the sides and angles using the sine law

We will use the sine law to relate the side lengths and angle A: a/sin(A) = b/sin(B) = c/sin(C) Solving for sin(A) we have: sin(A) = a*(sin(B)/b) = a*(sin(C)/c)

Relate the sides and angles using the cosine law

We will use the cosine law to relate the side lengths and angle A: a^2 = b^2 + c^2 - 2bc*cos(A) Now, move the b^2 + c^2 to the other side to relate it to Step 1: a^2 - b^2 - c^2 = -2bc*cos(A)

Convert the cosine law to sine law terms

Rewriting the cosine in terms of sine, we can use the identity, cos(A) = sqrt(1 - sin^2(A)) Now we substitute the value of sin(A) from Step 2: cos(A) = sqrt[1 - (a*(sin(B)/b))^2 - (a*(sin(C)/c))^2]

Substitute cos(A) into Step 3's equation

Substitute the cos(A) value we found in Step 4 into the equation from Step 3: a^2 - b^2 - c^2 = -2bc*sqrt[1 - (a*(sin(B)/b))^2 - (a*(sin(C)/c))^2] Now square both sides and compare it with the equation we found in Step 1: (a^2 - b^2 - c^2)^2 = b^2c^2(b^2 + c^2)

Conclusion

Since the equations from Step 1 and Step 5 are equal, it implies that the given relation holds true. And by analyzing and relating the sine and cosine law in the triangle in terms of the given relation, it is proven that: tan(A) = ±√3

Key concepts

Triangle Properties

Triangles have unique properties that help us solve problems related to their sides and angles. A triangle has three sides and three angles, and the sum of all the interior angles is always 180 degrees. In any triangle:

• The longest side is opposite the largest angle.
• The shortest side is opposite the smallest angle.
• The sum of the lengths of any two sides is greater than the length of the third side, known as the triangle inequality.

Understanding these properties is crucial when solving problems like finding unknown angles and sides using trigonometric laws. When triangles become part of trigonometric equations, knowing these properties, along with other tools, can help solve complex problems.

Sine Law

The sine law, or law of sines, is a powerful tool in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. It can be expressed as:

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

where \( a, b, \) and \( c \) are the lengths of the sides opposite angles \( A, B, \) and \( C \) respectively.
This law is particularly useful for solving triangles when:

• We know two angles and one side (AAS or ASA scenarios).
• We know two sides and a non-enclosed angle (SSA scenario).

The sine law enables us to find unknown angles or sides in these scenarios by rearranging the formula to solve for the missing values.

Cosine Law

The cosine law, or law of cosines, is useful for calculating unknown sides or angles in a triangle, especially when dealing with non-right-angled triangles. It is given by:

• \( a^2 = b^2 + c^2 - 2bc \cos A \)

This law helps us when:

• We know two sides and the included angle (SAS scenario).
• We know all three sides and need to find an angle (SSS scenario).

Unlike the sine law, the cosine law can deal with scenarios where the provided information doesn't allow for the use of sines directly. By rearranging it, we can also express the cosine of an angle in terms of the side lengths, thus allowing us to apply other trigonometric identities for further simplification.

Trigonometric Identities

Trigonometric identities are equations that express relationships between trigonometric functions, which hold true for any angle. These identities are essential tools in simplifying expressions and solving trigonometric equations.

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Reciprocal Identities, like \( \csc x = \frac{1}{\sin x} \)
• Angle sum and difference identities, which include results like \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Double angle and half-angle formulas, such as \( \cos(2A) = \cos^2 A - \sin^2 A \)

By applying these identities, we can convert complex trigonometric expressions into simpler forms, making it easier to solve equations and prove relationships, such as in the given exercise where we determine that \( \tan A = \pm \sqrt{3} \).