Question

If \(\sin \theta=\frac{1}{5},\) find \(\csc (\theta+\pi)\)

Short answer

\text{csc}(\theta + \pi) = -5

Step-by-step solution

Identify the Given Information

Given \(\text{sin}(\theta) = \frac{1}{5} \), identify that sin is provided for angle \( \theta \).

Recall the Definition of Cosecant

Recall that \( \text{csc}(x) = \frac{1}{\text{sin}(x)} \).

Use the Periodicity of Trigonometric Functions

Remember that \( \text{csc}(\theta + \pi) \) relates to the periodic properties of trigonometric functions. Specifically, \( \text{sin}(\theta + \pi) = -\text{sin}(\theta) \).

Substitute the Given Value

Substitute the given \( \text{sin}(\theta) = \frac{1}{5} \) into the equation: \( \text{sin}(\theta + \pi) = -\frac{1}{5} \).

Calculate the Required Cosecant

Using the definition of cosecant, compute \( \text{csc}(\theta + \pi) = \frac{1}{\text{sin}(\theta + \pi)} = \frac{1}{-\frac{1}{5}} = -5 \).

Key concepts

cosecant

The cosecant function, abbreviated as \(\text{csc}(x)\), is the reciprocal of the sine function. Remember, the sine function gives the ratio of the opposite side to the hypotenuse in a right triangle. Therefore: \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\).
So, if \(\text{sin}(x)\) is known, you can find \(\text{csc}(x)\) by simply taking its reciprocal.

This relationship allows us to solve problems involving sine and cosecant easily.
In the given problem, knowing that \(\text{sin}(\theta) = \frac{1}{5}\), we find the cosecant directly as \(\text{csc}(\theta) = \frac{1}{\frac{1}{5}} = 5\).

trigonometric periodicity

Trigonometric functions like sine and cosecant have periodic properties. This means their values repeat at regular intervals. For sine and cosecant, the period is \(\text{2}\text{π}\).
This periodicity helps in simplifying expressions involving shifted angles. For instance, the sine function has a property:

• \(\text{sin}(\theta + \text{π}) = -\text{sin}(\theta)\)

This relation tells us that if you shift the angle \(\theta\) by \(\text{π}\), the sine value simply becomes the negative of what it was.
Using this property in the problem, \(\text{sin}(\theta + \text{π}) = - \frac{1}{5}\).
Periodic properties make trigonometric problem-solving more straightforward, especially when dealing with angles beyond the typical range.

sine function

The sine function is fundamental in trigonometry. It describes a smooth, wave-like motion. In a right triangle, \(\text{sin}(x)\) gives the ratio of the length of the opposite side to the hypotenuse. The sine function has a range of [-1, 1] and is periodic with a period of \(\text{2}\text{π}\).
If \(\text{sin}(x)\) is known, other trigonometric functions can also be determined.

In the problem, we have \(\text{sin}(\theta) = \frac{1}{5}\). Since the sine function is periodic, \(\text{sin}(\theta + \text{π})\) can be calculated using the property:

• \(\text{sin}(\theta + \text{π}) = - \text{sin}(\theta) \)

Substituting the known value: \(\text{sin}(\theta + \text{π}) = - \frac{1}{5} \).
Understanding the sine function helps in various mathematical and real-world applications, including waves, oscillations, and circular motion.