Evaluate the expression. tan − 1 1 \tan^{-1}1 tan − 1 1

this problem, we're asked to evaluate the expression the inverse tangent of 1. Now here we can also think of this as okay the tangent of what angle is going to be equal to 1 and we're trying to find the angle for which that is true. Now I know that my angle has to be in between negative pi over 2 and positive pi over 2 because that is our specified interval when working with the inverse tangent. And I also know that because this is a positive number, positive one, I am specifically looking in this first quadrant where all of my tangent values are positive. So for which value in this or for which angle in this first quadrant is the tangent positive 1? Well for pi over 4 I see that my tangent is indeed 1 so that represents my solution pi over 4 for the inverse tangent of 1. Thanks for watching and I'll see you in the next one.

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
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5. Graphical Applications of Derivatives6h 2m
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7. Antiderivatives & Indefinite Integrals1h 26m
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10. Physics Applications of Integrals 3h 16m
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11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
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15. Power Series2h 19m
- Introduction to Power Series
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16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Inverse Trigonometric Functions

Calculus

0. Functions

Inverse Trigonometric Functions

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Multiple Choice

Evaluate the expression.
$\tan^{-1}1$

$0$

$pi over 4$

$pi over 2$

$- \frac{π}{4}$

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Verified step by step guidance

Understand that $ \tan^{-1}(x) $ is the inverse tangent function, which gives the angle whose tangent is $ x $.

Recognize that $ \tan^{-1}(1) $ asks for the angle $ \theta $ such that $ \tan(\theta) = 1 $.

Recall that the tangent of $ \frac{\pi}{4} $ is 1, because $ \tan(\frac{\pi}{4}) = \frac{\sin(\frac{\pi}{4})}{\cos(\frac{\pi}{4})} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $.

Consider the range of the inverse tangent function, which is $ (-\frac{\pi}{2}, \frac{\pi}{2}) $. Within this range, $ \frac{\pi}{4} $ is the angle that satisfies $ \tan(\theta) = 1 $.

Conclude that $ \tan^{-1}(1) = \frac{\pi}{4} $, as it is the angle within the range of the inverse tangent function that has a tangent of 1.