Problem 41 - Precalculus with Limits

Question

In Exercises 37 - 42, find the exact value of the expression. \( \dfrac{\tan\left(5\pi/6\right) - \tan \left(\pi/6\right)}{1 + \tan\left(5\pi/6\right) \tan \left(\pi/6\right)} \)

Step-by-Step Solution

Answer

The exact value of the expression is \( -\sqrt{3} \).

Step 1: Find the value of the tangent functions

First find the value of \( \tan(5\pi/6) \) and \( \tan(\pi/6) \). For \( \tan(5\pi/6) \) we know that \( \tan(\pi - \theta) = -\tan(\theta) \) by the properties of tangent function, so \( \tan(\pi/6) \) is \( \sqrt{3}/3 \). Therefore, \( \tan(5\pi/6) = -\tan(\pi/6) \) = \( -\sqrt{3}/3 \).

Step 2: Substitute the values into the expression

Substitute \( \tan(5\pi/6) \) with \( -\sqrt{3}/3 \) and \( \tan(\pi/6) \) with \( \sqrt{3}/3 \) to get \( \dfrac{-\sqrt{3}/3 - \sqrt{3}/3}{1 - \sqrt{3}/3 \times \sqrt{3}/3} \)

Step 3: Simplify the expression

After performing the operation, you'll get \( \dfrac{-2\sqrt{3}/3}{1 - 1/3} \) which simplifies further to \( -\sqrt{3} \).

Step 4: Final Answer

Therefore, the value of the expression is \( -\sqrt{3} \).

Key Concepts

tangent function properties

Understanding the properties of the tangent function is essential for solving trigonometric expressions effectively. The tangent function, often denoted as \( \tan(\theta) \), is a fundamental trigonometric function that relates the angle \( \theta \) to the ratio of the opposite side to the adjacent side in a right triangle.

One notable property of the tangent function is its periodicity. The function is periodic with a period of \( \pi \), meaning that \( \tan(\theta + \pi) = \tan(\theta) \). This property can be particularly useful when dealing with angles that are not standard, such as those given in radians.

Also, remember that the tangent function has symmetries. For example, \( \tan(\pi - \theta) = -\tan(\theta) \). This identity is crucial because it allows us to determine the tangent of an angle like \( 5\pi/6 \) by recognizing that \( \tan(5\pi/6) = -\tan(\pi/6) \).

• \( \tan(\pi/6) = \sqrt{3}/3 \)
• \( \tan(5\pi/6) = -\tan(\pi/6) = -\sqrt{3}/3 \)

These properties empower us to transform trigonometric expressions into their simplest forms by using known values.

radian measure

Radian measure is a way of expressing angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians offer a more mathematical understanding of angles based on arc length.

One radian is defined as the angle created when the arc length is equal to the radius of the circle. There are \( 2\pi \) radians in a full circle, which equals \( 360 \) degrees. This makes the conversion particularly straightforward: \( \pi \) radians is equal to \( 180^{\circ} \).

For the problems that deal with angles like \( 5\pi/6 \) and \( \pi/6 \), knowing the radian measure helps simplify finding the trigonometric functions. It's important to realize how radians correspond to well-known angles in degrees:

• \( \pi/6 \) radians = \( 30^{\circ} \)
• \( 5\pi/6 \) radians = \( 150^{\circ} \)

Understanding radian measure is critical for applying trigonometric identities effectively, especially in non-standard angle settings like those encountered in the exercise.

simplifying expressions

Simplifying trigonometric expressions requires a solid grasp of identities and mathematical operations. Starting with known values is a critical first step in this process.

For the expression \( \dfrac{\tan(5\pi/6) - \tan(\pi/6)}{1 + \tan(5\pi/6) \tan(\pi/6)} \), the first step is to plug in the values found for \( \tan(5\pi/6) \) and \( \tan(\pi/6) \).

• \( \tan(5\pi/6) = -\sqrt{3}/3 \)
• \( \tan(\pi/6) = \sqrt{3}/3 \)

Substituting these into the expression gives: \( \dfrac{-\sqrt{3}/3 - \sqrt{3}/3}{1 - \sqrt{3}/3 \times \sqrt{3}/3} \).

Next, perform the arithmetic operations:

• The numerator simplifies to \( -2\sqrt{3}/3 \).
• The denominator simplifies to \( 1 - 1/3 = 2/3 \).

Finally, dividing \(-2\sqrt{3}/3 \) by \( 2/3 \) leaves us with \(-\sqrt{3} \). Always remember that simplifying expressions helps in recognizing the neat pattern that trigonometric identities are known for.