Question

Find the exact value of \(\tan 40^{\circ}+\tan 140^{\circ} .\)

Short answer

0

Step-by-step solution

- Recall the tangent addition formula

For any two angles, \(A\) and \(B\), the tangent addition formula is \[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] This will be useful for simplifying our problem.

- Use tangent properties for specific angles

Recall that for any angle \(A\), \[ \tan(180^{\circ} - A) = -\tan A \] Applying this property, we have: \[ \tan 140^{\circ} = \tan(180^{\circ} - 40^{\circ}) = -\tan 40^{\circ} \]

- Substitute and simplify

Using the values from above, substitute into the expression: \[ \tan 40^{\circ} + \tan 140^{\circ} = \tan 40^{\circ} + (-\tan 40^{\circ}) = 0 \] Therefore, \[ \tan 40^{\circ} + \tan 140^{\circ} = 0 \]

Key concepts

tangent of an angle

The tangent of an angle in a right triangle is a trigonometric function that represents the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, it's expressed as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] where \(\theta\) is the angle in question. This basic definition is foundational in trigonometry and is used extensively to solve various problems.
For example, the tangent function is periodic, meaning it repeats its values at regular intervals. Specifically, \(\tan(\theta + 180^{\circ}) = \tan(\theta)\). This property is crucial when dealing with angles outside the standard range of \(0^{\circ}\) to \(90^{\circ}\).
In the problem provided, we used the property that \( \tan(180^{\circ} - A) = -\tan(A) \) to find the tangent of 140 degrees. Understanding such properties can simplify many calculations and make complex problems more manageable.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable within their domains. These identities are essential tools in simplifying and solving trigonometric problems. Here are some key identities:

• Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle sum and difference identities:
  \( \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \)
  \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)
• Double angle formulas:
  \( \sin(2A) = 2\sin(A)\cos(A) \)
  \( \cos(2A) = \cos^2(A) - \sin^2(A) \)

For our exercise, the tangent addition formula is crucial: \[ \tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)} \] This formula allows us to express the tangent of a sum of two angles in terms of the tangents of the individual angles. In the case of tangent subtraction, it changes slightly: \[ \tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)} \] Understanding these identities helps in breaking down complicated expressions, making them easier to work with.

sum of tangents

The sum of tangents of two angles can also be simplified using known trigonometric properties and identities. In the given problem, we had to find the value of \( \tan(40^{\circ}) + \tan(140^{\circ}) \). Here's a step-by-step reasoning:
First, recall the property: \( \tan(180^{\circ} - A) = -\tan(A) \). This allows us to rewrite \( \tan(140^{\circ}) \) as \( -\tan(40^{\circ}) \).
So, the expression \( \tan(40^{\circ}) + \tan(140^{\circ}) \) becomes \( \tan(40^{\circ}) + (-\tan(40^{\circ})) \), which simplifies to 0.
This simplification comes from understanding how the tangent of supplementary angles behaves. These supplementary angle properties are very useful for solving many trigonometric problems quickly.