cos^(-1)(4/3) + tan^(-1)(3/3) = tan^(-1)(3/4) + tan^(-1)(3/3)
Understand the Problem
The question appears to be asking to solve the trigonometric equation involving inverse cosine and inverse tangent functions. It involves the relationships between these functions and likely requires step-by-step mathematical manipulation to verify or solve the equation.
Answer
The left side is undefined; the right side is $ \tan^{-1}(3/4) + \frac{\pi}{4} $.
Answer for screen readers
The left side is undefined, while the right side equals $ \tan^{-1}(3/4) + \frac{\pi}{4} $.
Steps to Solve
- Determine Inverse Cosine Value
Check the term $ \cos^{-1}(4/3) $. Since the cosine function outputs values between -1 and 1, $ 4/3 $ is not in this range. Therefore, $ \cos^{-1}(4/3) $ is undefined.
- Evaluate Inverse Tangent Term
Next, evaluate $ \tan^{-1}(3/3) = \tan^{-1}(1) $. The value of this is:
$$ \tan^{-1}(1) = \frac{\pi}{4} $$
- Simplify Right Side of the Equation
Now, we look at the right side:
$$ \tan^{-1}(3/4) + \tan^{-1}(3/3) = \tan^{-1}(3/4) + \frac{\pi}{4} $$
- Combine the Results
Since we found that the left side is undefined, while the right side is defined, we have:
$$ \text{Left Side: } \text{undefined} $$ $$ \text{Right Side: } \tan^{-1}(3/4) + \frac{\pi}{4} $$
Therefore, the equation does not hold.
The left side is undefined, while the right side equals $ \tan^{-1}(3/4) + \frac{\pi}{4} $.
More Information
The term $ \cos^{-1}(4/3) $ is undefined because the cosine function can only take values between -1 and 1. Inverse tangent, however, can take wider ranges of real numbers.
Tips
- Assuming that $ \cos^{-1}(4/3) $ can produce a real number value, which it cannot.
- Mixing up the properties of inverse trigonometric functions.
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