Problem 20 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve: \(\sin ^{6} x+\cos ^{6} x=\frac{7}{16}\)

Step-by-Step Solution

Answer

After solving quadratic equation for cosine, the roots for the quadratic equation would give the square of the cosine of \(x\), taking the square root of the roots will give the cosine values from which the actual values of \(x\) can be determined considering the limits on the quadrant for cosine.

Step 1: Identifying and Using Pythagorean Identity

Rewrite the equation \(\sin ^{6} x+\cos ^{6} x=\frac{7}{16}\) as \((\sin ^{2} x)^{3}+(\cos ^{2} x)^{3}=\frac{7}{16}\)

Step 2: Applying Cube Sum Formulas

Rewrite \((\sin ^{2} x)^{3}+(\cos ^{2} x)^{3}\) as \((\sin^{2}x + \cos^{2}x)((\sin^{2}x + \cos^{2}x)^2 - 3\sin^{2}x\cos^{2}x)\) using cube sum formulae. Considering the identity \(\sin^{2}x + \cos^{2}x = 1\) we can further simplify the equation to \(1*(1 - 3\sin^{2}x\cos^{2}x) = \frac{7}{16}\)

Step 3: Rewriting in terms of \(\sin^{2}x\) or \(\cos^{2}x\)

Using the identity \(\sin^{2}x = 1 - \cos^{2}x\), the equation can be further simplified. Replace \(\sin^{2}x\) in the equation \(1*(1 - 3\sin^{2}x\cos^{2}x) = \frac{7}{16}\) to get \(1 - 3(1 - \cos^{2}x)\cos^{2}x = \frac{7}{16}\).

Step 4: Solving Quadratic Equation for Cosine

Expand and rearrange the equation to get it in standard quadratic form: \(3\cos^{4}x - 3\cos^{2}x + \frac{9}{16} = 0\). Solve this equation for \(\cos^{2}x\).

Step 5: Take square root and apply cosine values

By taking the square root of the value obtained for \(\cos^{2}x\), we get \(cos x\) and applying the symmetrical properties of the cosine function, we will get all the values of \(x\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry. It states that for any angle \(x\), \[ \sin^2 x + \cos^2 x = 1 \] This identity is derived from the Pythagorean theorem in geometry, relating the sides of a right-angled triangle. In our original exercise, we utilize this identity to simplify trigonometric expressions.
When solving the equation \(\sin^6 x + \cos^6 x = \frac{7}{16}\), we first rewrite it as \((\sin^2 x)^3 + (\cos^2 x)^3\). This highlights the importance of the Pythagorean Identity, as it allows us to substitute \(\sin^2 x + \cos^2 x\) with 1, and thus, simplify the expression. We see its practical utility in turning a potentially complex trigonometric equation into a more manageable form. Understanding this identity can make solving similar problems much easier.

Cube Sum Formula

The Cube Sum Formula is particularly useful when dealing with cube terms in equations. It states that for any two numbers \(a\) and \(b\), \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] We used this formula in our trigonometric exercise to further simplify the expression \((\sin^2 x)^3 + (\cos^2 x)^3\).
By applying the Cube Sum Formula, the expression can be rewritten as \[ (\sin^2 x + \cos^2 x)((\sin^2 x)^2 - \sin^2 x \cos^2x + (\cos^2 x)^2) \] Knowing that \( \sin^2 x + \cos^2 x = 1 \), this further reduces to: \[ 1((1 - 3\sin^2 x \cos^2 x)) \] This simplification is pivotal in managing and solving the original equation. A clear understanding of the cube sum formula aids in transforming potential hurdles into straightforward paths to the solution by approaching cubes strategically.

Quadratic Equation in Trigonometry

When dealing with trigonometric expressions, we often encounter quadratic equations. In the context of the trigonometric exercise, we were led to a quadratic equation form by replacing \(\sin^2 x\) with \(1 - \cos^2 x\).
Our equation simplifies to:\[ 3\cos^4 x - 3\cos^2 x + \frac{9}{16} = 0 \] This is a standard form of a quadratic equation in terms of \(\cos^2 x\). Solving quadratic equations involves:

• Rearranging the equation into standard form: \(ax^2 + bx + c = 0\).
• Applying the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In this exercise, we solve for \(\cos^2 x\), and further steps involve taking the square root to find \(\cos x\). The solution involves considering the symmetry of the cosine function to find all possible values for \(x\). Understanding how to handle quadratic equations such as these is essential in trigonometry, as they frequently appear in various problems.