Question

Evaluate the function without using a calculator. \(\csc \left(-420^{\circ}\right)\)

Short answer

\(\csc(-420^{\circ}) = -\frac{2\sqrt{3}}{3}\).

Step-by-step solution

Simplification

Since the sine function has a periodicity of \(360^{\circ}\), the desired angle is simplified by adding \(360^{\circ}\) to \(-420^{\circ}\), getting \(-60^{\circ}\).

Using Sin's properties

Use the property of sine, \(\sin(-\theta) = -\sin(\theta)\). So, \(\sin(-60^{\circ}) = -\sin(60^{\circ})\).

Evaluate Sine

Now, \(\sin(60^{\circ})\) is a well-known value, it is \(\frac{\sqrt{3}}{2}\). So, \(\sin(-60^{\circ}) = -\frac{\sqrt{3}}{2}\).

Evaluate Cosecant

Finally, knowing that \(\csc(\theta) = \frac{1}{\sin(\theta)}\), then \(\csc(-420^{\circ}) = \frac{1}{\sin(-420^{\circ})} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}\). Convert this into a rational number by multiplying the numerator and the denominator by \(\sqrt{3}\), therefore \(\csc(-420^{\circ}) = -\frac{2\sqrt{3}}{3}\).

Key concepts

Sine Function

The sine function is one of the fundamental trigonometric functions used to relate angles in a triangle to the sides. It is denoted as \( \sin(\theta) \), where \( \theta \) represents the angle in question. The sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This can be written as:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

When we talk about sine in terms of angles greater than 90°, we extend our understanding using the unit circle, which helps in understanding sine for any angle from 0° to 360° and beyond. For example, the sine of 60° is a known value: \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \).
In this exercise, understanding the sine function is crucial as it is the starting point to find the cosecant of the angle given. By calculating the sine and then finding its reciprocal, we can easily determine the value of the cosecant function.

Cosecant Function

The cosecant function is a reciprocal trigonometric function, which is denoted as \( \csc(\theta) \). It is the reciprocal of the sine function, defined as:

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

This function is important in scenarios where the length of the hypotenuse is known, and the opposite side needs to be calculated.
For angles where the sine function is easily determined, the cosecant is just the flip of this value. For example, if \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \), then \( \csc(60^{\circ}) = \frac{2}{\sqrt{3}} \).
When evaluated correctly, using the properties of sine, as we've done in our solution for \( \csc(-420^{\circ})\), you'll notice that simplification and rationalization are key steps to achieving an exact answer.

Angle Simplification

In trigonometry, angles can be simplified to find equivalent angles that have the same trigonometric function values. This process utilizes the periodic nature of trigonometric functions. The sine and cosine functions have a period of \( 360^{\circ} \), meaning values repeat every full rotation around the circle.
For negative angles or those beyond 360°, we can add or subtract 360° until we bring the angle into a range such as \( 0^{\circ} \) to \( 360^{\circ} \). In this problem, \( -420^{\circ} \) was simplified by adding 360° to get \( -60^{\circ} \). These two angles produce the same sine and cosecant values: they are equivalent in terms of their trigonometric outcomes.
This simplification is particularly helpful in trigonometry as it reduces complexity and allows us to apply known values and identities, making calculations easier.

Trigonometric Identities

Trigonometric identities are mathematical properties that establish relationships between trigonometric functions. These identities help simplify complex expressions and prove equalities in geometric problems.

• One key identity used in this exercise is \( \sin(-\theta) = -\sin(\theta) \). This tells us that sine is an odd function, demonstrating its symmetry about the origin.
• Another crucial identity is the reciprocal identity for cosecant: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

These identities are particularly helpful in reducing angles and understanding their behaviors, as shown in this exercise. By leveraging \( \sin(-60^{\circ}) = -\sin(60^{\circ}) \), simplification becomes clear, and finding reciprocal values (cosecant) follows naturally.
Mastering these identities aids in solving a wide array of trigonometric problems with confidence and accuracy.