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Chapter 9: Problem 17

Show \(\tan \left(360^{\circ}-\theta\right)=-\tan \theta\)

Short Answer

Expert verified
Question: Show that \(\tan \left(360^{\circ}-\theta\right)=-\tan \theta\). Answer: We used the tangent subtraction formula, converted degrees to radians, and applied the tangent periodicity property to demonstrate that \(\tan \left(360^{\circ}-\theta\right)=-\tan \theta\).

Step by step solution

01

Understand the given expression.

We have to show that \(\tan \left(360^{\circ}-\theta\right)=-\tan \theta\). This means finding a connection between the two sides of the equation using trigonometric properties and identities.
02

Convert degrees to radians.

To work with trigonometric functions, we typically convert angles in degrees to radians. To do this, we use the conversion factor \(\pi = 180^{\circ}\). Therefore, \(360^{\circ} = 2\pi\) radians, and the expression becomes \(\tan \left(2\pi - \theta\right) = -\tan \theta\).
03

Use the tangent subtraction formula.

We can apply the tangent subtraction formula: \(\tan \left( a - b \right) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\) In our case, \(a = 2\pi\) and \(b = \theta\). Applying the formula, we get: \(\tan \left( 2 \pi - \theta \right) = \frac{\tan (2\pi) - \tan \theta}{1 + \tan (2\pi) \tan \theta}\)
04

Use the tangent periodicity property.

The tangent function has a periodicity of \(\pi\), meaning \(\tan (x + n\pi) = \tan x\) for any integer \(n\). In our case, \(\tan 2 \pi = \tan 0 = 0\), as \(2\pi\) is an integer multiple of \(\pi\). Substitute this into our equation: \(\tan \left( 2 \pi - \theta \right) = \frac{0 - \tan \theta}{1 + 0 \cdot \tan \theta }\)
05

Simplify and arrive at the result.

Now, we can simplify the equation: \(\tan \left( 2 \pi - \theta \right) = \frac{-\tan \theta}{1}\) This simplifies to: \(\tan \left( 2 \pi - \theta \right) = -\tan \theta\) So we have shown that \(\tan \left( 360^{\circ}- \theta \right) = -\tan \theta\).

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