Question

The equation a \(\sin x+b \cos x=c\) where \(|c|>\sqrt{a^{2}+b^{2}}\) has (a) one solution (b) two solutions (c) no solution (d) infinite number of solutions The minimum value of \((\sin \alpha+\operatorname{cosec} \alpha)^{2}+(\cos \alpha+\sec \alpha)^{2}\) is

Short answer

Question: Determine the number of solutions for the equation \(|c|>\sqrt{a^{2}+b^{2}}\) and find the minimum value of the given expression \((\sin \alpha+\operatorname{cosec} \alpha)^{2}+(\cos \alpha+\operatorname{sec} \alpha)^{2}\). Answer: The equation \(|c|>\sqrt{a^{2}+b^{2}}\) has no solutions. The minimum value of the given expression \((\sin \alpha+\operatorname{cosec} \alpha)^{2}+(\cos \alpha+\operatorname{sec} \alpha)^{2}\) is \(4\).

Step-by-step solution

Rewrite inequality

Divide both sides of the inequality by \(\sqrt{a^{2}+b^{2}}\): $$\frac{|c|}{\sqrt{a^{2}+b^{2}}}>1$$

Apply transformation

Take the square root of \(a^{2}+b^{2}\) as the hypotenuse of a triangle with side lengths "\(a\)" and "\(b\)". Let \(\alpha = \sin^{-1}(\frac{a}{\sqrt{a^{2}+b^{2}}})\) and \(\beta = \cos^{-1}(\frac{b}{\sqrt{a^{2}+b^{2}}})\). Now, the inequality becomes \(\left|\sin(\alpha)\sin(x)+\cos(\beta)\cos(x)\right|>1\)

Trigonometric identity

Use the identity \(\sin(p+q)=\sin p\cos q+\cos p\sin q\) to rewrite the inequality as: $$|\sin(\alpha+\beta)|>1$$

Determine the number of solutions

Recall that the sine function has a range of \([-1,1]\). Since \(|\sin(\alpha+\beta)|\) is always non-negative, and the inequality states that \(|\sin(\alpha+\beta)|>1\), there is no solution to this inequality. Therefore, the equation has no solution, which corresponds to option (c). ##Part 2: Minimum value##

Simplify expression

Using the Pythagorean identities and algebra, we can simplify the given expression as follows: $$\begin{aligned} (\sin \alpha+\operatorname{cosec} \alpha)^{2}+(\cos \alpha+\sec \alpha)^{2}&=\left(\sin\alpha+\frac{1}{\sin\alpha}\right)^{2}+\left(\cos\alpha+\frac{1}{\cos\alpha}\right)^{2}\\ &=\left(\frac{\sin^2\alpha+2+\frac{1}{\sin^2\alpha}}{\sin^2\alpha}\right)+\left(\frac{\cos^2\alpha+2+\frac{1}{\cos^2\alpha}}{\cos^2\alpha}\right)\\ &=\frac{1+\sin^2\alpha}{\sin^2\alpha}+\frac{1+\cos^2\alpha}{\cos^2\alpha}\\ &=\frac{1+\cos^2\alpha}{1-\sin^2\alpha}+\frac{1+\sin^2\alpha}{1-\cos^2\alpha}\end{aligned}$$

Apply AM-GM inequality

Using the AM-GM inequality, we know that for any non-negative numbers \(p, q\), \(\frac{p+q}{2}\ge \sqrt{pq}\). Apply this inequality to the given simplified expression: $$\frac{1+\cos^2\alpha}{1-\sin^2\alpha}+\frac{1+\sin^2\alpha}{1-\cos^2\alpha}\ge 2\sqrt{\frac{(1+\cos^2\alpha)(1+\sin^2\alpha)}{(1-\sin^2\alpha)(1-\cos^2\alpha)}}$$ Since the denominator is equal to \(\sin^2\alpha\cos^2\alpha\), then: $$\frac{1+\cos^2\alpha}{1-\sin^2\alpha}+\frac{1+\sin^2\alpha}{1-\cos^2\alpha}\ge 2\sqrt{\frac{(1+\cos^2\alpha)(1+\sin^2\alpha)}{\sin^2\alpha\cos^2\alpha}}\ge 2\sqrt{\frac{4}{\sin^2\alpha\cos^2\alpha}}=4$$ Equality occurs when \(\sin^2\alpha=\cos^2\alpha=\frac{1}{\sqrt{2}}\). Therefore, the minimum value of \((\sin \alpha+\operatorname{cosec} \alpha)^{2}+(\cos \alpha+\sec \alpha)^{2}\) is \(4\).

Key concepts

Inequalities in Trigonometry

When it comes to tackling trigonometric equations, understanding inequalities is crucial. Inequalities in trigonometry reveal the relationship between the sizes of different trigonometric values. An easy way to think about these is to visualize trigonometric functions as waves that oscillate between certain bounds; for sine and cosine, these bounds are \textbf{-1 and 1}.

For the given problem of the equation \(a \sin x + b \cos x = c\), we find that no solution exists if \(|c| > \sqrt{a^2 + b^2}\). This is because the inequality suggests that the sum of the sine and cosine function amplitudes must be larger than the maximum possible value, which is beyond the allowable range of the sine function. The visualization of sine as a wave is particularly useful here; since it can never reach beyond 1 or -1, any requirement to do so renders the equation unsolvable. This concept helps students understand the boundaries within which trigonometric solutions must fall.

AM-GM Inequality

The AM-GM inequality is an essential concept in mathematics, notably in optimizing problems or finding minimum and maximum values. Standing for Arithmetic Mean-Geometric Mean inequality, it essentially states that the arithmetic mean of a set of non-negative numbers is greater than or equal to the geometric mean of the same set. In simpler terms, for non-negative \(a\) and \(b\), \(\frac{a + b}{2} \geq \sqrt{ab}\).

In the context of finding the minimum value of \((\sin \alpha + \operatorname{cosec} \alpha)^2 + (\cos \alpha + \sec \alpha)^2\), application of the AM-GM inequality simplifies the problem. By restructuring the expression using Pythagorean identities, you can reveal a form suitable for the AM-GM inequality. After applying this powerful tool, the inequality shows that the minimum value of this trigonometric expression is 4. This example beautifully illustrates how inequalities can be used to set bounds on trigonometric expressions and underscores why the AM-GM inequality is such a valuable tool in mathematical problem-solving.

Pythagorean Identities

Pythagorean identities form the backbone of many trigonometric solutions and simplifications. These identities are derived from the Pythagorean theorem, which describes the fundamental relationship between the sides of a right-angled triangle. The most common Pythagorean identities involve sine and cosine functions:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• \(1 + \tan^2(\theta) = \sec^2(\theta)\)
• \(1 + \cot^2(\theta) = \csc^2(\theta)\)

Applying these identities allows us to rewrite trigonometric expressions in more manageable forms. In our case, simplifying the expression \((\sin \alpha + \operatorname{cosec} \alpha)^2 + (\cos \alpha + \sec \alpha)^2\) hinged on transforming it into a format that made it amenable to the application of the AM-GM inequality. By doing so we could establish the minimum value of the original expression. Recognizing when and how to apply Pythagorean identities is a crucial skill in solving trigonometric equations and can greatly clarify what might initially appear to be a complex problem.