Problem 37 - Algebra 2

Question

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \csc 70^{\circ} $$

Step-by-Step Solution

Answer

The approximate value of \(\csc 70^{\circ}\) to four decimal places is 1.0642.

Step 1: Convert Degrees to Radians

The trigonometric functions in calculators usually use radians instead of degrees. The conversion from degrees to radians is given by \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Thus, convert 70 degrees to radians: \(70^{\circ} = 70 \times \frac{\pi}{180}\) radians.

Step 2: Compute the Sine of the Angle

After converting the angle to radians, compute the sine of the converted angle using a calculator: \(\sin(70 \times \frac{\pi}{180})\). Note that the sin function refers to the sine of an angle.

Step 3: Compute the Reciprocal

As the cosecant function is the reciprocal of the sine function, compute the inverse of the sine value obtained in Step 2 to get the value of the cosecant function: \(\csc 70^{\circ} = 1 / \sin(70 \times \frac{\pi}{180})\). Round this to four decimal places as required.

Key Concepts

Radians to Degrees Conversion

The conversion between radians and degrees is crucial when working with trigonometric functions, especially with calculators. The standard formula to convert an angle from degrees to radians is \[\text{radians} = \text{degrees} \times \frac{\pi}{180}.\] This formula helps you switch between the two scales efficiently.
For example, to convert 70 degrees to radians, you multiply 70 by \( \frac{\pi}{180} \), which equals \( 70 \times \frac{\pi}{180} \approx 1.2217 \text{ radians} \).
This step is necessary because most advanced calculations and calculators use radians as the default angle measurement unit.

Cosecant Function

The cosecant function, \( \csc(\theta) \), represents the reciprocal of the sine function. It's an important component in trigonometry, often used to solve problems involving right triangles and oscillations.
Given by the formula:\[\csc(\theta) = \frac{1}{\sin(\theta)},\]it is undefined when the sine of the angle, \( \sin(\theta) \), becomes zero since division by zero is not possible.
For example, calculating \( \csc 70^{\circ} \) involves first determining \( \sin 70^{\circ} \), then taking its reciprocal. This gives you insight into the ratio of the hypotenuse to the opposite side in a right triangle.

Trigonometric Calculations

Trigonometric calculations form the foundation of various mathematical and engineering computations. These calculations are essential for solving angles, lengths, and other spatial parameters.
To perform these calculations effectively, understanding the relationship between different trigonometric functions, like sine and cosecant, is key.

• Sine (\( \sin \)) relates an angle to the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosecant (\( \csc \)) is the reciprocal of sine, which means \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Always ensure that your angles are correctly converted into radians when using a calculator, as this prevents errors in your computations.
Knowing these connections allows for solving complex problems in physics, architecture, and other applied sciences comfortably.