What is the value of log 10 (cos θ) + log 10 (sin θ) + log 10 (tan θ) + log 10 (cot θ) + log 10 (sec θ) + log 10 (cosesc θ)?
0
Solving Logarithms with Trigonometric Functions
The problem asks for the value of a sum of logarithms with base 10, where the arguments are trigonometric functions of \(\theta\). To solve this, we will use properties of logarithms and trigonometric identities.
Understanding the Problem Statement
We need to calculate the value of:
\(\log_{10}(\cos \theta) + \log_{10}(\sin \theta) + \log_{10}(\tan \theta) + \log_{10}(\cot \theta) + \log_{10}(\sec \theta) + \log_{10}(\csc \theta)\)
Note: The term 'cosesc \(\theta\)' is interpreted as 'cosecant \(\theta\)' or 'csc \(\theta\)'.
Applying Logarithm Properties
A key property of logarithms states that the sum of logarithms with the same base is the logarithm of the product of their arguments:
\(\log_b(x) + \log_b(y) = \log_b(xy)\)
Using this property, we can combine the given expression into a single logarithm:
\(\log_{10}(\cos \theta \cdot \sin \theta \cdot \tan \theta \cdot \cot \theta \cdot \sec \theta \cdot \csc \theta)\)
Using Trigonometric Identities
Now, we will express the tangent, cotangent, secant, and cosecant functions in terms of sine and cosine:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\csc \theta = \frac{1}{\sin \theta}\)
Substituting Identities and Simplifying the Product
Substitute these identities into the product inside the logarithm:
\(\cos \theta \cdot \sin \theta \cdot \left(\frac{\sin \theta}{\cos \theta}\right) \cdot \left(\frac{\cos \theta}{\sin \theta}\right) \cdot \left(\frac{1}{\cos \theta}\right) \cdot \left(\frac{1}{\sin \theta}\right)\)
Let's simplify this product. We can rearrange the terms and cancel where possible:
\(\left(\cos \theta \cdot \frac{1}{\cos \theta}\right) \cdot \left(\sin \theta \cdot \frac{1}{\sin \theta}\right) \cdot \left(\frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta}\right)\)
Each pair of terms multiplies to 1 (assuming \(\sin \theta \neq 0\) and \(\cos \theta \neq 0\), which must be true for the original logarithms to be defined):
- \(\cos \theta \cdot \frac{1}{\cos \theta} = 1\)
- \(\sin \theta \cdot \frac{1}{\sin \theta} = 1\)
- \(\frac{\sin \theta}{\cos \theta} \cdot \frac{\cos \theta}{\sin \theta} = 1\)
So, the product inside the logarithm simplifies to:
\(1 \cdot 1 \cdot 1 = 1\)
Evaluating the Final Logarithm
The original expression simplifies to \(\log_{10}(1)\).
The logarithm of 1 to any base \(b > 0, b \neq 1\) is always 0.
\(\log_{10}(1) = 0\)
Conclusion
The value of the given expression is 0.
Let's check the options:
| Option | Value |
|---|---|
| 1 | -1 |
| 2 | 0 |
| 3 | 0.5 |
| 4 | 1 |
The calculated value, 0, matches Option 2.
Revision Table: Logarithm and Trigonometry Review
| Concept | Property/Identity | Example |
|---|---|---|
| Sum of Logs | \(\log_b x + \log_b y = \log_b (xy)\) | \(\log_{10} 2 + \log_{10} 5 = \log_{10} (2 \times 5) = \log_{10} 10 = 1\) |
| Log of 1 | \(\log_b 1 = 0\) (for \(b > 0, b \neq 1\)) | \(\log_e 1 = 0\), \(\log_{100} 1 = 0\) |
| Tangent Identity | \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) | If \(\sin \theta = 1/2\), \(\cos \theta = \sqrt{3}/2\), then \(\tan \theta = (1/2) / (\sqrt{3}/2) = 1/\sqrt{3}\) |
| Secant Identity | \(\sec \theta = \frac{1}{\cos \theta}\) | If \(\cos \theta = 1/2\), then \(\sec \theta = 1 / (1/2) = 2\) |
| Cosecant Identity | \(\csc \theta = \frac{1}{\sin \theta}\) | If \(\sin \theta = \sqrt{3}/2\), then \(\csc \theta = 1 / (\sqrt{3}/2) = 2/\sqrt{3}\) |
| Cotangent Identity | \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\) | If \(\tan \theta = 1\), then \(\cot \theta = 1/1 = 1\) |
Additional Information on Logarithm and Trigonometric Functions
For the logarithms \(\log_{10}(\cos \theta)\), \(\log_{10}(\sin \theta)\), etc., to be defined, the arguments must be positive. This means \(\cos \theta > 0\), \(\sin \theta > 0\), \(\tan \theta > 0\), etc.
The conditions \(\cos \theta > 0\) and \(\sin \theta > 0\) together imply that \(\theta\) must be in the first quadrant (excluding the axes), i.e., \(0 < \theta < \pi/2\) radians or \(0^\circ < \theta < 90^\circ\). In this quadrant, all basic trigonometric functions (\(\sin \theta\), \(\cos \theta\), \(\tan \theta\), \(\cot \theta\), \(\sec \theta\), \(\csc \theta\)) are positive, ensuring their logarithms are real numbers.
The property \(\log_b 1 = 0\) is a fundamental property of logarithms. It comes directly from the definition of a logarithm: \(\log_b x = y\) means \(b^y = x\). So, \(\log_b 1 = y\) means \(b^y = 1\). For any base \(b > 0, b \neq 1\), the only value of \(y\) that satisfies this equation is \(y=0\).
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