Question

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

Short answer

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

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.

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 \(\tan \left(2\pi - \theta\right) = -\tan \theta\).

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: \(\tan \left( 2 \pi - \theta \right) = \frac{\tan (2\pi) - \tan \theta}{1 + \tan (2\pi) \tan \theta}\)

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: \(\tan \left( 2 \pi - \theta \right) = \frac{0 - \tan \theta}{1 + 0 \cdot \tan \theta }\)

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

Key concepts

Tangent Subtraction Formula

The tangent subtraction formula is a powerful tool used in trigonometry to evaluate the tangent of a difference between two angles. The formula is expressed as:

• \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \)

When working with this formula, it is key to identify the values of \(a\) and \(b\) in your problem and substitute them into the equation. In the case of the original exercise, we set \( a = 2\pi \) and \( b = \theta \). By applying the formula, we observe how the angles are manipulated utilizing the identity. This formula simplifies complex expressions and is especially handy when angles are expressed in terms of radians or when simplifying trigonometric equations.

Degrees to Radians Conversion

Converting angles from degrees to radians is a necessary step in many trigonometric analyses. Degrees and radians are both units that measure angles, but radians provide a more natural measure when working with trigonometric functions. To convert degrees to radians, use the conversion factor \( \pi = 180^{\circ} \). Therefore, to convert an angle such as \( 360^{\circ} \),we write it as:

• \( 360^{\circ} \times \frac{\pi}{180^{\circ}} = 2\pi \text{ radians} \)

This conversion is crucial to properly use trigonometric properties and formulas, as these typically require angles to be expressed in radians. This is why in the original exercise, \(360^{\circ}\) is converted to \(2\pi\) radians before applying further trigonometric identities.

Periodicity of Trigonometric Functions

One interesting property of trigonometric functions like tangent is their periodicity. The tangent function, \( \tan x \), is periodic with a period of \( \pi \). This means that:

• \( \tan(x + n\pi) = \tan x \)

where \( n \) is any integer. This property enables simplified computation across numerous situations, especially where angles add up to multiples of \( \pi \).In the original step-by-step solution, applying periodicity was crucial when finding that \( \tan 2\pi = \tan 0 = 0 \) because \( 2\pi \) is a full rotation equivalent to \( 0 \),implying no change in the original value of \( \tan \) for those inputs. Recognizing and applying periodicity helps simplify the process of solving more complex trigonometric expressions.