Question

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \csc \frac{5 \pi}{13} $$

Short answer

The approximate value of \( \text{csc} \frac{5 \pi}{13} \) is 1.09.

Step-by-step solution

- Understand the Function

The function given is \(\text{csc} \frac{5 \pi}{13}\), which stands for the cosecant. Cosecant is the reciprocal of the sine function. So, \(\text{csc} x = \frac{1}{\text{sin} x}\).

- Evaluate the Argument

The argument inside the cosecant function is \(\frac{5 \pi}{13}\). This value needs to be used in the sine function calculation.

- Use a Calculator to Find Sine

Use a calculator to find the sine of \(\frac{5 \pi}{13}\). Make sure your calculator is set to radian mode. The value of \(\text{sin} \frac{5 \pi}{13}\) is approximately 0.9205.

- Find the Cosecant

Now, find the reciprocal of the sine value. \(\text{csc} \frac{5 \pi}{13} = \frac{1}{\text{sin} \frac{5 \pi}{13}}\). So, \(\text{csc} \frac{5 \pi}{13} = \frac{1}{0.9205}\).

- Divide and Round

Divide 1 by 0.9205 to find the approximate value. \(\frac{1}{0.9205}\) gives approximately 1.0863. Round this to two decimal places to get 1.09.

Key concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics and describe relationships between the angles and sides of triangles. There are six main trigonometric functions:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Cosecant (csc)
• Secant (sec)
• Cotangent (cot)

These functions are frequently used in various scientific fields, including physics, engineering, and computer science. Among them, sine, cosine, and tangent are the primary functions, while cosecant, secant, and cotangent are their reciprocals, respectively.

Radian Mode

Radians are a way of measuring angles in terms of the radius of a circle. One complete rotation around a circle is equal to 2π radians. Unlike degrees, which divide a circle into 360 parts, radians measure angles as a fraction of the circle's circumference.
When calculating trigonometric functions, it's crucial to set your calculator to the correct mode: radian mode or degree mode. For example, in the exercise involving \(\text{csc} \frac{5 \pi}{13}\), we use radian mode. Ensure your calculator is set to radian mode before performing the calculations to avoid incorrect results.

Reciprocal Identities

Reciprocal identities relate the six trigonometric functions to each other. They are helpful for simplifying complex trigonometric expressions.
Key reciprocal identities include:

• \(\text{csc} x = \frac{1}{\text{sin} x}\)
• \(\text{sec} x = \frac{1}{\text{cos} x}\)
• \(\text{cot} x = \frac{1}{\text{tan} x}\)

In the exercise, we use the identity \(\text{csc} x = \frac{1}{\text{sin} x}\) to find \(\text{csc} \frac{5 \pi}{13}\). Understanding these identities allows you to switch between different trigonometric functions easily.

Sine Function

The sine function is one of the fundamental trigonometric functions. It relates a given angle to the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Mathematically, it is represented as:
\( \text{sin} x = \frac{\text{opposite}}{\text{hypotenuse}} \)
The sine function has a periodic nature with a principal range of [-1, 1] and a period of 2π radians.
In the given exercise, we find the sine of an angle \(\frac{5 \pi}{13}\) in radian mode using a calculator, which yields an approximate value of 0.9205. We then use this to find the cosecant by taking its reciprocal, which further demonstrates the practical use of the sine function in solving more complex trigonometric problems.