Consider the following statements: 1. The value of cos 61° + sin 29° cannot exceed 1. 2. The value of tan 23° - cot 67° is less than 0. Which of the above statements is / are correct?
1 only
Evaluating Trigonometric Statements
Let's analyze each statement given in the question involving trigonometric expressions. We will use trigonometric identities and properties to determine if each statement is correct.
Statement 1: Value of cos 61° + sin 29° cannot exceed 1.
The first statement claims that the value of \(\cos 61^\circ + \sin 29^\circ\) is less than or equal to 1. To verify this, we can use the complementary angle identity.
We know that \(\sin \theta = \cos (90^\circ - \theta)\).
Using this identity for the term \(\sin 29^\circ\):
\(\sin 29^\circ = \sin (90^\circ - 61^\circ)\)
\(\sin 29^\circ = \cos 61^\circ\)
Now, substitute this back into the original expression:
\(\cos 61^\circ + \sin 29^\circ = \cos 61^\circ + \cos 61^\circ = 2 \cos 61^\circ\)
So, the statement is equivalent to checking if \(2 \cos 61^\circ \le 1\). This can be simplified to checking if \(\cos 61^\circ \le 0.5\).
We know the exact value of \(\cos 60^\circ\) is 0.5.
\(\cos 60^\circ = 0.5\)
In the range from \(0^\circ\) to \(90^\circ\), the cosine function is a decreasing function. This means that for angles greater than \(60^\circ\) but less than or equal to \(90^\circ\), the cosine value will be less than \(\cos 60^\circ\).
Since \(61^\circ > 60^\circ\), it follows that \(\cos 61^\circ < \cos 60^\circ\).
Therefore, \(\cos 61^\circ < 0.5\).
Multiplying both sides by 2 (a positive number), the inequality direction remains the same:
\(2 \cos 61^\circ < 2 \times 0.5\)
\(2 \cos 61^\circ < 1\)
Since \(2 \cos 61^\circ\) is strictly less than 1, it certainly cannot exceed 1. Thus, Statement 1 is correct.
Statement 2: Value of tan 23° - cot 67° is less than 0.
The second statement claims that the value of \(\tan 23^\circ - \cot 67^\circ\) is negative (less than 0). We can use the complementary angle identity here as well.
We know that \(\cot \theta = \tan (90^\circ - \theta)\).
Using this identity for the term \(\cot 67^\circ\):
\(\cot 67^\circ = \cot (90^\circ - 23^\circ)\)
\(\cot 67^\circ = \tan 23^\circ\)
Now, substitute this back into the original expression:
\(\tan 23^\circ - \cot 67^\circ = \tan 23^\circ - \tan 23^\circ\)
\(\tan 23^\circ - \cot 67^\circ = 0\)
The statement claims that the value is less than 0. Our calculation shows the value is exactly 0.
Since 0 is not less than 0, Statement 2 is incorrect.
Conclusion on the Statements
Based on our analysis:
- Statement 1: The value of \(\cos 61^\circ + \sin 29^\circ\) cannot exceed 1. This is correct.
- Statement 2: The value of \(\tan 23^\circ - \cot 67^\circ\) is less than 0. This is incorrect.
Therefore, only Statement 1 is correct.
Revision Table: Trigonometric Identities Used
| Identity | Description | Used in Statement |
|---|---|---|
| \(\sin (90^\circ - \theta) = \cos \theta\) | Sine of a complementary angle is cosine of the angle. | Statement 1 |
| \(\cot (90^\circ - \theta) = \tan \theta\) | Cotangent of a complementary angle is tangent of the angle. | Statement 2 |
Additional Information: Properties of Trigonometric Functions
Understanding the behavior of trigonometric functions in different quadrants is crucial for solving such problems.
- For angles \(\theta\) between \(0^\circ\) and \(90^\circ\):
- Both \(\sin \theta\) and \(\cos \theta\) are positive.
- Both \(\tan \theta\) and \(\cot \theta\) are positive.
- \(\sin \theta\) is an increasing function (as \(\theta\) increases, \(\sin \theta\) increases).
- \(\cos \theta\) is a decreasing function (as \(\theta\) increases, \(\cos \theta\) decreases).
- \(\tan \theta\) is an increasing function.
- \(\cot \theta\) is a decreasing function.
In Statement 1, we used the decreasing property of \(\cos \theta\) in the \(0^\circ\) to \(90^\circ\) range to compare \(\cos 61^\circ\) with \(\cos 60^\circ\). Since \(61^\circ > 60^\circ\), and cosine is decreasing, \(\cos 61^\circ < \cos 60^\circ\).
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