Problem 15
Question
For any real \(x\), the maximum value of \(\cos ^{2}(\cos x)+\sin ^{2}(\sin x)\) is (a) 1 (b) \(1+\sin ^{2} 1\) (c) \(1+\cos ^{2} 1\) (d) None.
Step-by-Step Solution
Verified Answer
The maximum value of the given expression is \(1+\sin ^{2} 1\), so option (b) is the correct answer.
1Step 1: Analyze the Given Expression
The expression given is \(\cos ^{2}(\cos x)+\sin ^{2}(\sin x)\). Trigonometric functions like cosine and sine take values between -1 and 1. When these values are squared, the result is always in the range 0 to 1. So, the values of \(\cos ^{2}(\cos x)\) and \(\sin ^{2}(\sin x)\) will each fall in this range, between 0 and 1.
2Step 2: Calculate Maximum Values
The maximum value of each term will be obtained when \(\cos x = 1\) and \(\sin x = 1\). Therefore \(\cos ^{2}(1) = 1\) and \(\sin ^{2}(1) \approx 0.708\). Adding these together will give a result just above 1.
3Step 3: Select the Correct Answer
Looking at the options, the only possible answer that is slightly more than 1 is option (b) which is \(1+\sin ^{2} 1\). This matches the value we have computed.
Key Concepts
Cosine and Sine ValuesTrigonometric IdentitiesIIT JEE Trigonometry
Cosine and Sine Values
Understanding cosine and sine values is a basic yet essential aspect of trigonometry. These values range from -1 to 1, inclusive. These functions are periodic, which means they repeat their values in regular intervals. Given a real number x, both cos(x) and sin(x) will output a value within this range.
When you square these values, as seen in the exercise, the squared value, either \( \cos^2(x) \) or \( \sin^2(x) \) will always fall between 0 and 1. This is because squaring a number makes it positive, and since the original value of cosine or sine is never more than 1, the square cannot exceed 1 either. Recognizing this constraint is crucial in solving problems where you're asked to find maximum or minimum values of trigonometric expressions.
When you square these values, as seen in the exercise, the squared value, either \( \cos^2(x) \) or \( \sin^2(x) \) will always fall between 0 and 1. This is because squaring a number makes it positive, and since the original value of cosine or sine is never more than 1, the square cannot exceed 1 either. Recognizing this constraint is crucial in solving problems where you're asked to find maximum or minimum values of trigonometric expressions.
Trigonometric Identities
We use trigonometric identities to simplify and solve trigonometry problems. These identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the equation are defined.
One of the fundamental identities is the Pythagorean identity: \(\cos^2(x) + \sin^2(x) = 1\). It arises from the Pythagorean theorem applied to a right triangle. However, in our exercise, we encounter a modified version with nested functions, which isn’t a standard identity but still can be approached using the concept of range for trigonometric functions.
As the cosine and sine functions oscillate between -1 and 1, any identities involving these functions will also be constrained by these values. This becomes particularly important when distinguishing between possible maximum or minimum values in expressions.
One of the fundamental identities is the Pythagorean identity: \(\cos^2(x) + \sin^2(x) = 1\). It arises from the Pythagorean theorem applied to a right triangle. However, in our exercise, we encounter a modified version with nested functions, which isn’t a standard identity but still can be approached using the concept of range for trigonometric functions.
As the cosine and sine functions oscillate between -1 and 1, any identities involving these functions will also be constrained by these values. This becomes particularly important when distinguishing between possible maximum or minimum values in expressions.
IIT JEE Trigonometry
Preparing for competitive exams like the IIT JEE (Indian Institutes of Technology Joint Entrance Examination) requires robust understanding and application of trigonometry. Trigonometry in IIT JEE often includes complex problems that test a student's grasp on trigonometric functions, identities, and their maxima and minima.
The solution provided follows the logical approach necessary for such competitive exams. First, it considers the range of sine and cosine functions. Then it evaluates the maximum possible values of those functions' squares, to finally conclude with the correct solution. Students preparing for exams must not only practice step-by-step solutions but also strengthen their conceptual understanding to tackle variations of standard problems, such as maximizing or minimizing an expression involving trigonometric values and identities.
The solution provided follows the logical approach necessary for such competitive exams. First, it considers the range of sine and cosine functions. Then it evaluates the maximum possible values of those functions' squares, to finally conclude with the correct solution. Students preparing for exams must not only practice step-by-step solutions but also strengthen their conceptual understanding to tackle variations of standard problems, such as maximizing or minimizing an expression involving trigonometric values and identities.
Other exercises in this chapter
Problem 14
If \(\cos 25^{\circ}+\sin 25^{\circ}=p\), then find \(\cos 50^{\circ}\).
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If \(A+B+C=\pi\), then prove that \(\cot A+\cot B+\cot C-\operatorname{cosec} A \operatorname{cosec} B \operatorname{cosec} C\) \(=\cot A \cdot \cot B \cdot \co
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Find the maximum value of \(12 \sin \theta-9 \sin ^{2} \theta\).
View solution Problem 16
If \(\tan \alpha=\frac{1}{\sqrt{x\left(x^{2}+x+1\right)}} \tan \beta=\frac{\sqrt{x}}{\sqrt{\left(x^{2}+x+1\right)}}\) and \(\tan \gamma=\frac{\sqrt{\left(x^{2}+
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