Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \csc 60^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \csc 60^{\circ} \) is \( \frac{2\sqrt{3}}{3} \).

Step 1: Recall Definition of Cosecant

The cosecant of an angle is the reciprocal of the sine of that angle. Mathematically, this is expressed as \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Identify the Known Sine Value

We know from trigonometric tables or the unit circle that \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \).

Step 3: Calculate the Cosecant

Substitute the sine value into the cosecant formula: \( \csc 60^{\circ} = \frac{1}{\sin 60^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2}} \).

Step 4: Simplify the Reciprocal

Simplify the expression for cosecant. Taking the reciprocal gives: \( \csc 60^{\circ} = \frac{2}{\sqrt{3}} \).

Step 5: Rationalize the Denominator

To rationalize the denominator, multiply numerator and denominator by \( \sqrt{3} \): \[ \csc 60^{\circ} = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \].

Key Concepts

Trigonometric Ratios

Trigonometric ratios are fundamental elements in trigonometry, a branch of mathematics focused on the relationships between the angles and sides of triangles. These ratios include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent.
For a given angle \(\theta\), these primary trigonometric ratios are defined as follows:

• \(\sin \theta\) (Sine) = Opposite Side/Hypotenuse
• \(\cos \theta\) (Cosine) = Adjacent Side/Hypotenuse
• \(\tan \theta\) (Tangent) = Opposite Side/Adjacent Side

Each of these has a corresponding reciprocal function:

• \(\csc \theta\) (Cosecant) = 1/\(\sin \theta\)
• \(\sec \theta\) (Secant) = 1/\(\cos \theta\)
• \(\cot \theta\) (Cotangent) = 1/\(\tan \theta\)

Understanding these ratios is essential, as they are widely used to solve various trigonometric problems, such as calculating angles and side lengths in right triangles.

Angle Measures

Angle measures can be expressed in degrees or radians. The most familiar unit is degrees, where a full circle equates to 360 degrees. This can be split into familiar segments:

• 90 degrees for a right angle
• 180 degrees for a straight angle
• 60 degrees, which is often found in equilateral triangles

When working with triangles and trigonometric functions, knowing key angles and their sine, cosine, and tangent values can significantly aid problem-solving.
For example, some of the most common trigonometric angle measures include:

• \(\sin 30^{\circ} = \frac{1}{2}\)
• \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)
• \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\)

These known values simplify calculations involving their reciprocal functions, such as solving for cosecant, where \(\csc 60^{\circ} = \frac{1}{\sin 60^{\circ}}\). Understanding these foundational angles is key to mastering trigonometry.

Rationalizing Denominators

Rationalizing denominators is a technique used to eliminate irrational numbers from the denominator of a fraction. This is particularly common when dealing with square roots. The aim is to transform the expression into a form that is often easier to understand or use.
To rationalize a denominator containing a square root, multiply both the numerator and the denominator by the same square root. This process eliminates the square root from the denominator:

• Original: \(\frac{2}{\sqrt{3}}\)
• Multiply by: \(\frac{\sqrt{3}}{\sqrt{3}}\)
• = \(\frac{2\sqrt{3}}{3}\)

Once rationalized, expressions are considered to be in their simplest form. This simplification step is crucial when working with trigonometric functions, making the problems easier to manage and solutions more elegant.