Question

Show \(\tan \left(180^{\circ}-\theta\right)=-\tan \theta\)

Short answer

Question: Prove that \(\tan \left(180^{\circ}-\theta\right) = -\tan \theta\). Solution: To prove the identity, we first converted the angle measurement from degrees to radians. This allowed us to apply the formula for the tangent of the difference of two angles. By evaluating the tangent of pi and substituting it into the expression, we were able to simplify and show that \(\tan \left(180^{\circ}-\theta\right)=-\tan \theta\).

Step-by-step solution

Convert degrees to radians

To make it easier to work with this expression, we will first convert the angle measurement from degrees to radians. Recall that \(180^{\circ}=\pi\) radians. So, we can rewrite the given expression as: $$ \tan \left( \pi - \theta \right) = -\tan \theta $$

Use tangent of angle difference formula

Recall the formula for the tangent of the difference of two angles: $$ \tan (A-B) = \frac{\tan A - \tan B}{1 + \tan A\tan B} $$ In our case, \(A=\pi\) and \(B=\theta\). Let's apply this formula to our expression: $$ \tan \left(\pi - \theta\right) = \frac{\tan \pi - \tan \theta}{1 + \tan \pi\tan \theta} $$

Evaluate tangent of pi

Recall that the tangent of an angle is the ratio of the sine to the cosine: \(\tan x = \frac{\sin x}{\cos x}\). Let's evaluate \(\tan \pi\): $$ \tan \pi =\frac{ \sin \pi}{\cos \pi} = \frac{0}{-1} = 0 $$

Substitute tangent of pi in the expression

Now that we have the value of \(\tan \pi\), let's substitute it into the expression from Step 2: $$ \tan \left(\pi - \theta\right) = \frac{0 - \tan \theta}{1 + 0\tan \theta} = \frac{-\tan \theta}{1} $$

Simplify and conclude the proof

Finally, we can simplify our expression to obtain the desired result: $$ \tan \left(\pi - \theta\right) = -\tan \theta $$ This completes our proof, and we've shown that \(\tan \left(180^{\circ}-\theta\right)=-\tan \theta\).

Key concepts

Angle Conversion

Converting angles is essential in trigonometry, especially when moving between degrees and radians. These two units measure angles in different ways, and understanding their relationship is crucial for solving many trigonometric problems.

• Degrees: This is a common unit for measuring angles, more intuitive in everyday life. A full circle is 360 degrees.
• Radians: This unit is based on the radius of a circle. One full circle is measured as \(2\pi\) radians.

To convert from degrees to radians, we use the fact that \(180^{\circ}=\pi\) radians. Therefore, to convert an angle from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\). For example, the conversion of \(180^{\circ}\) to radians results in \(\pi\). This foundational step makes many trigonometric calculations more manageable.

Tangent Function

The tangent function, denoted as \(\tan\theta\), is one of the fundamental trigonometric functions. It relates the angles of a right triangle to the ratios of two of its sides. Specifically, the tangent of an angle \(\theta\) is the ratio of the opposite side to the adjacent side.

• In a unit circle, the tangent of an angle can also be interpreted as the length of the segment where the terminal side of the angle intersects the line \(x = 1\).
• The tangent function is periodic with a period of \(\pi\), meaning \(\tan(\theta + \pi) = \tan\theta\).

In trigonometric identities, understanding the behavior of the tangent function helps us manipulate proofs, such as showing \(\tan(\pi - \theta) = -\tan\theta\), using properties such as symmetry and functional transformations.

Radian Measure

Radian measure is integral to advanced trigonometry. When an angle is expressed in radians, it often simplifies mathematical operations and expressions, especially within calculus and trigonometry.

• A radian is defined based on the radius of a circle. One radian is the angle created when the arc length equals the circle's radius.
• Radians allow for a uniform approach in mathematical functions, as many functions and forms in mathematics naturally extend using radian measure.

Utilizing radians becomes particularly useful when using calculus as it harmonizes with the unit circle approach, with \(\sin\) and \(\cos\) functions being defined by the circle's circumference. Notably, key points or angles in radians simplify trigonometric identities and proofs, translating complex degree-based identities into more accessible forms using unit circle properties.

Proof Verification

Verifying a trigonometric identity involves demonstrating that both sides of the equation represent the same quantity for all permissible angle values. It's like showing that two different paths lead to the same destination.

• Start by expressing all trigonometric functions in terms of sine and cosine, as they are the fundamental trigonometric functions.
• Simplify using known identities, such as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), and angle addition and subtraction formulas.

For our example, understanding that \(\tan(\pi - \theta)\) can be simplified using symmetry and properties of tangent allows the correct transformation to \(-\tan\theta\). This shows continuity and logical flow in proving identities. The completion of a proof provides confidence that no matter the angle \(\theta\), the identity holds true, ensuring mathematical consistency.