Question

Cotangent, Secant, and Cosecant by Calculator. Evaluate to four decimal places. $$\csc 207.4^{\circ}$$

Short answer

Use a calculator to find \( \csc 207.4^\circ \) by first converting the angle to radians, then calculating the sine, taking its reciprocal, and rounding the result to four decimal places.

Step-by-step solution

Understand the Cosecant Function

The cosecant function is the reciprocal of the sine function. So, to find \( \csc(\theta) \), you calculate \( \frac{1}{\sin(\theta)} \) where \(\theta\) is the angle in degrees.

Convert Angle to Radians

Most calculators require the angle to be in radians for trigonometric functions. To convert 207.4 degrees to radians, use the conversion factor \(\frac{\pi}{180}\). The calculation is \( 207.4 \cdot \frac{\pi}{180} \) radians.

Calculate Sine of the Angle

Use your calculator to find \( \sin(207.4^\circ) \) or \( \sin \) of the angle in radians from Step 2.

Determine Cosecant

Take the reciprocal of the sine value from Step 3 to find the cosecant of the angle. Ensure your calculator is set to the correct mode (degrees or radians) as determined in steps 1 and 2.

Round the Result

Round the calculated cosecant value to four decimal places as required.

Key concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in the study of mathematics, especially when dealing with right-angled triangles and the properties of waves. In essence, they relate the angles of a triangle to the ratios of its sides. Among the primary trigonometric functions are sine, cosine, and tangent, which are often abbreviated as sin, cos, and tan, respectively.

The function we are focusing on today is the cosecant, denoted as \(\csc\), which is the reciprocal of the sine function \(\sin\). In other words, for any angle \(\theta\), \(\csc(\theta) = \frac{1}{\sin(\theta)}\). It's crucial when working with cosecant to remember that it's undefined when sine is zero because you cannot divide by zero.

Trigonometric functions come into play in various practical applications such as physics, engineering, astronomy, and even in complex financial models. They are also essential for solving problems involving periodic phenomena, such as sound waves and light.

Converting Degrees to Radians

When working with trigonometric functions, you'll often need to convert angles from degrees to radians, since radians are the standard unit of angular measure in mathematics. This is particularly important because many calculators and computer programs use radians for trigonometric calculations.

To convert degrees to radians, use the conversion factor \(\pi\ radians / 180^{\circ}\). The conversion stems from the fact that a full circle is 360 degrees, which is equivalent to \(2\pi\) radians. So, for an angle \(A\) in degrees, the formula to convert to radians is \(A \cdot \frac{\pi}{180}\).

For example, to convert 207.4 degrees to radians, the calculation would be \(207.4^\circ \cdot \frac{\pi}{180}\), resulting in approximately 3.6224 radians (rounded to four decimal places). It's a simple multiplication but is crucial for ensuring accuracy in your calculations.

Inverse Sine Function

The inverse sine function, commonly known as arcsine and denoted as \(\sin^{-1}\) or \(\text{asin}\), is the inverse operation of the sine function. It allows you to determine an angle when you know the sine's value.

When you input the value of a sine into the inverse sine function, it yields the measure of the angle (in radians or degrees) whose sine is that value. It's important to note that since \(\sin\) is not a one-to-one function because it repeats its values over the interval \(0, 2\pi\), its inverse is defined only for the range \(\text{[-1, 1]}\). This means that the output of \(\sin^{-1}\) will always be within \(\text{[-}\frac{\pi}{2}, \frac{\pi}{2}\text{]}\) in radians or \(\text{[-90}^\circ, \text{90}^\circ\text{]}\) in degrees.

This function becomes particularly useful when you need to solve for an angle and you're given a trigonometric ratio. For example, if you're given that the sine of an angle is 0.5, you can find the angle by calculating \(\sin^{-1}(0.5)\), which would give you \(30^\circ\) or \(\frac{\pi}{6}\) radians.