Evaluate the expressions:
a) arctan(-sqrt(3))
a) Arctan(-sqrt(3))
So, I know how to find the answers with a calculator, but how do I find it step by step?
by test time you need to know the "standard" angles with easy-to-recall trig functions
0,π/6,π/4,π/3,π/2
If you know those angles and their trig ratios, you will recall that
tan π/3 = √3
Now, recall the bit about principal values of inverse trig functions. You restrict the domain so that the range is one continuous period, containing 0, if possible.
-π/2 <= Arcsin(x) < π/2
0 <= Arccos(x) <= π
-π/2 < Arctan(x) < π/2
...
arctan(-√3) = -π/3 + 2kπ
Arctan(-√3) = -π/3
Aah, okay! That's easy enough! That makes total sense. Thanks!
To find the angle whose tangent is equal to -sqrt(3), you can use the inverse tangent function, commonly denoted as arctan or tan^(-1). Here is how you can evaluate the expression step by step:
Step 1: Start with the given expression, arctan(-sqrt(3)).
Step 2: Recognize that arctan is the inverse of the tangent function, so you're essentially looking for the angle whose tangent is -sqrt(3).
Step 3: Recall that the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. Specifically, tan(theta) = opposite/adjacent.
Step 4: Since you know that the tangent of the angle you're looking for is -sqrt(3), you can set up the equation tan(theta) = -sqrt(3).
Step 5: To solve for theta, take the inverse tangent of both sides of the equation: arctan(tan(theta)) = arctan(-sqrt(3)).
Step 6: The inverse tangent cancels out the tangent function on the left side, leaving you with theta = arctan(-sqrt(3)).
Step 7: At this point, you would typically use a calculator or mathematical software to find the numerical value of arctan(-sqrt(3)), which is approximately -60 degrees or -π/3 radians.
So, to summarize, the expression arctan(-sqrt(3)) represents the angle whose tangent is equal to -sqrt(3), which is approximately -60 degrees or -π/3 radians.
To evaluate the expression arctan(-sqrt(3)) or Arctan(-sqrt(3)) step by step, we need to understand that it involves finding the inverse tangent or arctangent of the given value.
Here's how you can proceed:
Step 1: Calculate or determine the principal angle:
To find the inverse tangent of a value, we look for the angle whose tangent is equal to that value. However, since the tangent function is periodic, it has multiple angles that have the same tangent value. To determine the principal angle, we use the range of the arctangent function, which is -π/2 to π/2 or -90° to 90°.
Step 2: Calculate the value of arctan(-sqrt(3)):
The given value for the inverse tangent is -sqrt(3). To find the principal angle, we need to solve the equation:
tan(x) = -sqrt(3)
Since we are looking for an angle within the range of -90° to 90° or -π/2 to π/2, we know that the answer lies in the fourth quadrant (Q4) or the second quadrant (Q2).
In Q4, the tangent function is negative, so we can write:
tan(x) = -sqrt(3)
Combining this with the principal angle concept, we find x = arctan(-sqrt(3)), where x lies between -90° and 0° or -π/2 and 0.
Step 3: Determine the exact value:
To find the exact value of arctan(-sqrt(3)), we can use the reference triangle for 30° in Q4.
In a 30-60-90 triangle, the sides have the following ratio:
opposite side/hypotenuse = sqrt(3)/2
Using this ratio, we can determine the exact value:
tan(theta) = opposite side/adjacent side
tan(theta) = (sqrt(3)/2) / (1/2)
tan(theta) = sqrt(3)
Since tan(x) = -sqrt(3), we conclude that x = -30° or -π/6. Therefore, arctan(-sqrt(3)) equals -30° or -π/6.
Remember that angles in the opposite quadrants can also have the same tangent value, so further adjustment may be needed depending on the context of the problem you are solving.