# How do you show that inverse tan (1/x) = #pi/2-theta#, given that x is a positive number and #theta# = inverse tanx?

As explained below.

By signing up, you agree to our Terms of Service and Privacy Policy

To show that ( \text{arctan}\left(\frac{1}{x}\right) = \frac{\pi}{2} - \theta ), where ( x ) is a positive number and ( \theta = \text{arctan}(x) ), we use the properties of trigonometric functions.

Given that ( \theta = \text{arctan}(x) ), it means that ( \tan(\theta) = x ).

Now, let's consider ( \text{arctan}\left(\frac{1}{x}\right) ). By definition, this is the angle whose tangent is ( \frac{1}{x} ).

We can write this as ( \tan(\text{arctan}(1/x)) ).

Using the tangent of the difference formula, ( \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} ), we get:

( \tan(\text{arctan}(1/x) - \theta) = \frac{\tan(\text{arctan}(1/x)) - \tan(\theta)}{1 + \tan(\text{arctan}(1/x))\tan(\theta)} )

( = \frac{\frac{1}{x} - x}{1 + \frac{1}{x} \cdot x} )

( = \frac{\frac{1}{x} - x}{1 + 1} )

( = \frac{1 - x^2}{x(1 + x^2)} )

Now, ( \tan(\text{arctan}(1/x) - \theta) ) represents the tangent of the angle whose arctan is ( 1/x ) minus ( \theta ). Since ( x ) is positive, ( \theta ) is in the first quadrant, and ( \frac{\pi}{2} - \theta ) is in the second quadrant.

This implies that ( \tan(\frac{\pi}{2} - \theta) = \frac{1 - x^2}{x(1 + x^2)} ).

Therefore, ( \text{arctan}\left(\frac{1}{x}\right) = \frac{\pi}{2} - \theta ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you graph #r=2+4costheta# on a graphing utility?
- How do you convert #x^2 + y^2 = 4x# into polar form?
- Todd and Scott left the dining hall for a walk on two straight paths that diverge by 48°. Scott walked 580 m and Todd walked 940 m. How far apart are they?
- How do you divide # (6-10i) / (7-2i) #?
- What is the value of sin(-60°)?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7