If [tex]$\tan x = \frac{4}{3}$[/tex], what is the value of b?
Answer (2)
The given equation tan x = 3 4 leads to x ≈ 53.1 3 ∘ . However, the value of b is not clearly defined in the question, making it difficult to determine what b is without more context. Further clarification on the relationship between x and b is necessary for an accurate answer.
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The problem is poorly worded and likely contains a typo. Assuming the question meant to ask for the value of b where either x 4 = b and tan x = 3 4 , or x = b and tan x 4 = 3 4 , we find that the calculated values of b are not close to any of the provided options (4, 5, 6, 7). Therefore, it is impossible to determine the correct value of b from the given information. There is likely a typo in the question or the answer choices.
Explanation
Understanding the Problem We are given the equation tan x 4 = 3 4 and asked to find the value of b , where the possible values for b are 4, 5, 6, and 7. The problem is poorly worded, so we will consider two possible interpretations.
Case 1: x 4 = b and tan x = 4/3 Case 1: Assume the question meant to ask: If x 4 = b , what is the value of b if tan x = 3 4 ? In this case, we first find x by taking the arctangent of both sides of tan x = 3 4 to get x = arctan ( 3 4 ) . Then we calculate b = x 4 = ( arctan ( 3 4 ) ) 4 .
Calculating b in Case 1 Using a calculator, we find that arctan ( 3 4 ) ≈ 0.9273 . Therefore, b = ( 0.9273 ) 4 ≈ 0.7394 . Comparing this to the given options (4, 5, 6, 7), none of them are close to 0.7394.
Case 2: x = b and tan ( x 4 ) = 4/3 Case 2: Assume the question meant to ask: If x = b , what is the value of b if tan ( x 4 ) = 3 4 ? In this case, we first find x 4 by taking the arctangent of both sides of tan ( x 4 ) = 3 4 to get x 4 = arctan ( 3 4 ) . Then we take the fourth root of both sides to find x = ( arctan ( 3 4 ) ) 4 1 . Since x = b , then b = ( arctan ( 3 4 ) ) 4 1 .
Calculating b in Case 2 Using a calculator, we find that arctan ( 3 4 ) ≈ 0.9273 . Therefore, b = ( 0.9273 ) 4 1 ≈ 0.9813 . Comparing this to the given options (4, 5, 6, 7), none of them are close to 0.9813.
Reconsidering the Original Equation However, let's reconsider the original equation tan x 4 = 3 4 . If we assume that the question intended to ask for the value of x 4 , then x 4 = arctan ( 3 4 ) ≈ 0.9273 . Since none of the given options are close to this value, it is likely that there was a typo in the question.
Final Analysis Since none of the provided options (4, 5, 6, 7) are close to the calculated values in either case, it is likely that there is an error in the problem statement or the provided options. However, if we must choose the closest value from the given options, we can analyze the possibilities. In Case 1, b ≈ 0.7394 , which is far from any of the options. In Case 2, b ≈ 0.9813 , which is also far from any of the options. If the question was intended to ask for the value of x 4 , then x 4 ≈ 0.9273 , which is also far from any of the options. Given the options, it is not possible to determine the intended value of b .
Conclusion Without a clear relationship between x and b , and given that the calculated values are far from the provided options, it is impossible to determine the correct value of b . It is highly likely that there is a typo in the question. However, since we must choose from the given options, and given that arctan ( 4/3 ) ≈ 0.9273 , we can see if any of the options could be related to this value. If x 4 = b , then b = arctan ( 4/3 ) . However, none of the options are close to 0.9273. If x = b , then b = ( arctan ( 4/3 ) ) 1/4 ≈ 0.9813 . Again, none of the options are close to this value. Therefore, we cannot determine the value of b from the given information.
Examples
In trigonometry, understanding the relationships between angles and their tangent values is crucial for various applications, such as calculating heights and distances in surveying, navigation, and engineering. For example, if you know the angle of elevation to the top of a building and the distance to the building, you can use the tangent function to determine the height of the building. Similarly, in navigation, the tangent function can be used to calculate the course angle needed to reach a destination based on the distance and bearing.
