Problem 36 - Algebra 2

Question

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \csc 30^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \csc(30) \) is 2.

Step 1: Find the sine of the angle

The first step involves determining the sine of the given angle. For a 30 degrees angle, this could be done by recalling the standard trigonometric values. The sine of a 30 degrees angle is 0.5.

Step 2: Find the reciprocal of the sine

Next, as the cosecant (csc) is the reciprocal of the sine (sin), just take the reciprocal of the calculated sine value. Hence, \( \csc(30) = \frac{1}{\sin(30)} \)

Step 3: Calculate the reciprocal

Finally, calculate the reciprocal of 0.5 to get the value of \( \csc(30) \). Thus, \( \csc(30) = \frac{1}{0.5} = 2 \).

Key Concepts

Reciprocal Functions

Reciprocal functions are fascinating in the world of trigonometry. They come into play when we want to express one trigonometric function in terms of another.
In simple terms, the reciprocal of a function is just 1 divided by that function. For instance, if we have a function f(x), its reciprocal is 1/f(x).

• Reciprocal functions in trigonometry include cosecant (csc), secant (sec), and cotangent (cot).
• The cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.

Understanding reciprocal functions helps us solve various trigonometric problems involving these unique relationships. As in our example, knowing that \( \ ext{csc}(x) = \frac{1}{\sin(x)} \) is crucial for computations.

Cosecant

Cosecant, abbreviated as csc, is one of the less commonly used trigonometric functions compared to sine, cosine, and tangent.
However, it plays a vital role, especially when dealing with reciprocal identities.
The cosecant of an angle is the reciprocal of the sine of that angle.

• The formula to find cosecant of an angle θ is: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• This means if you know the sine value of an angle, calculating the cosecant is straightforward by taking its reciprocal.

For example, if \( \sin(30^{\circ}) = 0.5 \), then \( \csc(30^{\circ}) = \frac{1}{0.5} = 2 \). In trigonometry, understanding these relationships that derive from one another is essential for mastering the subject.

Trigonometric Values

Trigonometric values are specific outputs of trigonometric functions like sine, cosine, and tangent for different angles. These values are crucial for solving problems in trigonometry.
Much like our bodies rely on bones to stand, trigonometric values form the framework for working with trigonometric functions.

• Common angles often used are 0°, 30°, 45°, 60°, and 90°, each having particular sine, cosine, and tangent values.
• Knowing these values helps in easily finding reciprocals like cosecant, secant, and cotangent.

For example, a common trigonometric value is \( \sin(30^{\circ}) = 0.5 \). This is integral for calculations, as seen in our example, where it leads directly to \( \csc(30^{\circ}) = 2 \). Mastering these core values allows for more efficient problem-solving in trigonometry.