Question

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \tan \frac{8 \pi}{15} $$

Short answer

\( \tan \frac{8 \pi}{15} \approx -0.93 \)

Step-by-step solution

Identify the expression

The expression to evaluate is \( \tan \frac{8 \pi}{15} \).

Input the expression into the calculator

Enter \( \frac{8 \pi}{15} \) into the calculator to find the angle in radians. This specific input typically requires a calculator that supports the use of \( \pi \) or allows the input of the value of \( \pi \) directly.

Calculate the tangent

After entering \( \frac{8 \pi}{15} \), use the tangent function (\( \tan \)) button to find the value. Make sure the calculator is set to radians mode since the input angle is given in radians.

Round the result

The calculator will give an approximate value for \( \tan \frac{8 \pi}{15} \). This value needs to be rounded to two decimal places.

Key concepts

tangent function

In trigonometry, the tangent function (\tan) is one of the basic trigonometric functions used to relate angles to side lengths in a right-angled triangle.
For a given angle \theta, the tangent function is defined as the ratio of the opposite side to the adjacent side of the angle in the triangle.
This is expressed as:
\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
The tangent function is particularly useful when you need to find the slope or steepness of an angle. It is periodic with a period of \(\frac{\text{pi}}{2}\), meaning \(\tan(\theta + \frac{\text{pi}}{2}) = \tan(\theta)\).
It's important to remember that \(\tan(\theta)\) is defined for all real numbers except for \(\theta = \frac{\text{pi}}{2} + k\text{pi}\) where \(\text{k}\) is an integer. For these values, the function goes to infinity.

radians

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which are measured in parts of 360, radians are measured based on the circumference of the circle.
One full rotation around a circle is \(\text{2pi}\) radians, which corresponds to 360 degrees.
Therefore, \(\text{pi}\) radians equal 180 degrees. The relationship between degrees and radians allows for converting from one to the other using the conversion factor \(\frac{\text{pi}}{180}\) for degrees to radians and \(\frac{180}{\text{pi}}\) for radians to degrees.
When working with trigonometric functions like \(\tan\), it’s crucial to know which unit of measure you are using. Calculators typically have a setting that allows you to switch between degrees and radians. For example, to enter \(\frac{8\text{pi}}{15}\), ensure the calculator is set to radians mode.

using a calculator

When performing calculations with trigonometric functions, using a calculator is quite essential.
Here's a step-by-step approach to finding \(\tan \frac{8 \text{pi}}{15}\):

• First, make sure your calculator is in radians mode rather than degrees. This is usually done by pressing the MODE button and selecting 'RAD'.
  
• Input \(\frac{8 \text{pi}}{15}\) into the calculator. If your calculator supports the symbol \(\text{pi}\) directly, use it instead of the numerical value (3.14159).
  
• Once the angle is entered, press the tangent (\tan) button to get the value of \(\tan \frac{8 \text{pi}}{15}\).
  
• Finally, round the result to two decimal places for your final answer.
  

Using these steps will help you accurately and efficiently find values using trigonometric functions with a calculator.