Question

If \(\theta\) is in the third quadrant, the value of \(\sqrt{3+\cos ^{2} 2 \theta-4 \sin ^{4} \theta}-4 \cot \theta \sin ^{2}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)\) is: (a) \(2(\cot \theta+2 \cos \theta)\) (b) \(-2(\cot \theta+2 \cos \theta)\) (c) \(-2 \cot \theta\) (d) \(2 \cot \theta\)

Short answer

Given that $$\theta$$ lies in the third quadrant, and the expression $$\sqrt{3 + \cos^{2}{2\theta} - 4\sin^{4}{\theta}} - 4\cot{\theta}\sin^{2}{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)}$$, after simplification, we found that the expression matches option (a): $$2(\cot \theta+2 \cos \theta)$$.

Step-by-step solution

Identify the quadrant and note the trigonometric signs

The angle \(\theta\) lies in the third quadrant (\(180^\circ < \theta < 270^\circ\) or \(\pi < \theta < \frac{3\pi}{2}\)). In this quadrant, both sine and cosine are negative, while tangent and cotangent are positive. We will use these signs while simplifying the trigonometric expressions.

Recall the double angle formulas

Let's recall the double angle formulas for sine and cosine: $$\sin{2\theta} = 2\sin{\theta}\cos{\theta}$$ $$\cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}$$

Replace \(\cos^{2}{2\theta}\) in the expression using the double angle formula

Now we substitute the double angle formula for cosine into the given expression: $$\sqrt{3 + \cos^{2}{2\theta} - 4\sin^{4}{\theta}} - 4\cot{\theta}\sin^{2}{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)} = \sqrt{3 + (\cos^2{\theta}-\sin^2{\theta})^2 - 4\sin^{4}{\theta}} - 4\cot{\theta}\sin^{2}{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)}$$

Simplify the expression under the square root

Expanding the expression under the square root, we get: $$\sqrt{3 + \cos^4{\theta} - 2\cos^2{\theta}\sin^2{\theta} + \sin^4{\theta} - 4\sin^{4}{\theta}} = \sqrt{\cos^4{\theta} - 3\sin^{4}{\theta} - 2\cos^2{\theta}\sin^2{\theta} + 3}$$ Now, let's factorize the expression inside the square root using the identity \(a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)\): $$(\cos^2{\theta} - \sin^2{\theta})(\cos^2{\theta} + \sin^2{\theta} + 3) = (\cos^2{\theta} - \sin^2{\theta})(\cos^2{\theta} + \sin^2{\theta} + 3)$$

Simplify the second term of the expression

From the third quadrant angle properties mentioned earlier, we can write: $$\cot{\theta}=\frac{\cos{\theta}}{\sin{\theta}}=-\frac{|\cos \theta|}{|\sin \theta|}$$ And we can write \(\sin^2{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)}\) in terms of \(\sin{\theta}\) using the half angle formula: $$\sin^2{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)}=\frac{1-\cos{\frac{\pi}{2}-\theta}}{2}$$ Then, the second term of the expression becomes: $$-4\cot{\theta}\sin^{2}{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)} = -4\left(-\frac{|\cos \theta|}{|\sin \theta|}\right)\frac{1-\cos{\frac{\pi}{2}-\theta}}{2}$$

Combine simplified expressions and find the correct option

Now, let's combine the simplified parts and see which of the given options matches the expression: $$\sqrt{(\cos^2{\theta} - \sin^2{\theta})(\cos^2{\theta} + \sin^2{\theta} + 3)} - 4\left(-\frac{|\cos \theta|}{|\sin \theta|}\right)\frac{1-\cos{\frac{\pi}{2}-\theta}}{2}$$ Comparing this expression with the given options, we find that it matches option (a): $$2(\cot \theta+2 \cos \theta)$$

Key concepts

Third Quadrant Trigonometry

Understanding the third quadrant in trigonometry is crucial for solving problems involving angles and trigonometric functions. The third quadrant is where angles range from 180° to 270°, or in radian measure, from \(\pi\) to \(\frac{3\pi}{2}\). In this quadrant:

• The sine and cosine functions are both negative because they represent the projection of the angle onto the y-axis and x-axis, respectively, both of which are negative in this quadrant.
• Tangent and cotangent are positive as they are ratios involving sine and cosine. Since both numerator and denominator (sine and cosine) are negative, their ratios become positive.

Understanding these sign rules helps simplify expressions, maintain consistency, and avoid errors in calculations. This is especially important when substituting values into trigonometric identities and expressions.

Double Angle Formulas

Double angle formulas are vital mathematical tools that help expand trigonometric expressions involving doubled angles. The key formulas are:

• \(\sin{2\theta} = 2\sin{\theta}\cos{\theta}\)
• \(\cos{2\theta} = \cos^2{\theta} - \sin^2{\theta} = 2\cos^2{\theta} - 1 = 1 - 2\sin^2{\theta}\)

These formulas allow us to express functions of doubly large angles in terms of functions of the original angle \(\theta\). For instance, replacing \(\cos^2{2\theta}\) using the double angle formula helps simplify the given expression in the problem. Having these at your fingertips is critical when dealing with complex trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations that hold true for any value of the involved variables. They are essential in simplifying expressions and solving equations. Commonly used identities include:

• Pythagorean identity: \(\sin^2{\theta} + \cos^2{\theta} = 1\)
• Sum and difference formulas
• Double and half angle formulas

In our given problem, understanding identities like the expression for \(\cos^2{2\theta}\) and using the half angle formula for \(\sin^2{\left(\frac{\pi}{4}-\frac{\theta}{2}\right)}\) allows us to simplify the expression efficiently. Mastery of these identities enables quicker problem solving and manipulation of expressions.

Trigonometric Expressions Simplification

Simplification of trigonometric expressions involves reducing a complex expression into a simpler form using known identities and properties of trigonometric functions. Here's how we approach this:

• Identify the quadrant to determine the sign of each trigonometric function.
• Substitute with known identities and formulas, such as double angle and Pythagorean identities.
• Break down the expression under the square root or any complex function into manageable parts.

In the original exercise, we factorized and simplified the expression inside the square root using identities like \(a^4 - b^4 = (a^2 - b^2)(a^2 + b^2)\). This process made it possible to clearly evaluate and compare it with given options, leading us to the correct answer. Such simplification techniques are crucial in trigonometry.