Problem 26 - Algebra 2

Question

Evaluate each expression to the nearest hundredth. Each angle is given in radians. $$ \csc (-3.2) $$

Step-by-Step Solution

Answer

The value of \(csc(-3.2)\) to the nearest hundredth will depend on the exact value calculated by you using your calculator at each step. Once you've calculated it, round it to the nearest hundredth.

Step 1: Understand the cosecant function

The cosecant (csc) of an angle \(\theta\) is defined as the reciprocal of its sine, or \(csc( \theta ) = \frac{1}{\sin( \theta )}\). Hence, evaluating \(\csc (-3.2)\) essentially means calculating the reciprocal of \(\sin (-3.2)\).

Step 2: Calculate the sine of the negative angle

The sine of \(-3.2\) radian angle can be calculated using a scientific calculator. Most scientific calculators have buttons for trigonometric calculations and accept both degrees and radians as inputs. Make sure your calculator is set to radians, then just input \(-3.2\) and hit the 'sin' button.

Step 3: Calculate the reciprocal of the result

Once you've obtained the sine value, you then need to calculate its reciprocal to get the cosecant value. You can do this by dividing 1 by the sine value.

Step 4: Round the result

Finally, round the result you obtained to the nearest hundredth to get the final answer. This is typically done by looking at the third decimal place. If it's 5 or more, you round up the second decimal place, if it's less than 5, you keep it as it is.

Key Concepts

Cosecant Function

The cosecant function, represented as \( \csc(\theta) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. This means that for any angle \( \theta \),

• \( \csc(\theta) = \frac{1}{\sin(\theta)} \)

Understanding this relationship is crucial because it allows us to compute cosecant using basic sine functions. In practical terms, if you know the sine of an angle, then taking the reciprocal of that sine value gives you the cosecant. For negative angles, trigonometric identities like \( \sin(-\theta) = -\sin(\theta) \) can be used directly since the sine function is odd, affecting the result of the cosecant function as well.
Interestingly, the cosecant function is particularly useful in solving problems related to right triangles and wave functions in physics, where understanding the reciprocal nature helps in simplifying equations.

Radian Measure

Radian measure is a way of expressing angles based on the radius of a circle. It's a universal measure used in higher mathematics and physics. One radian corresponds to the angle made when the arc length is equal to the radius of the circle, and you can relate radians to degrees using the conversion factor:

• \( 180^\circ = \pi \text{ radians} \)
• \( 1 \text{ radian} \approx 57.2958^\circ \)

When working with trigonometric functions like cosecant, having angles in radians is common, because calculations are generally simpler in radians than in degrees, particularly in calculus.
Remember, in exercises, always ensure your calculator is set to radians if the angle is given in radians. This setting is crucial because it affects the computation of trigonometric functions directly.

Reciprocal Identities

Reciprocal identities are key relationships in trigonometry that relate trigonometric functions through reciprocals. These identities include:

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

They are useful in transforming complex trigonometric expressions into simpler forms and can aid in solving equations.
The essence of these identities lies in their ability to link one function with another, expanding the ways you can manipulate and understand angles. For instance, knowing that \( \csc(-3.2) \) requires computing \( \sin(-3.2) \), underscores how intertwined trigonometric functions are, and how reciprocal identities simplify the calculation process.

Rounding Decimal Numbers

Rounding decimal numbers is a fundamental skill necessary in mathematics to control precision and simplify results. In trigonometry, rounding often comes into play when expressing the results of trig functions. The method involves:

• Looking at the digit immediately after the digit you want to round to.
• If that digit is 5 or greater, you round the previous digit up.
• If it's less than 5, you leave the previous digit unchanged.

For example, when you compute \( \csc(-3.2) \), the result might be a long decimal. To express it "to the nearest hundredth," you consider the third decimal place.
Rounding helps not only in clarity but also when reporting results in scientific and engineering contexts where over-precision could be misleading or unnecessary. Being able to round appropriately ensures clear communication of numerical results.