Problem 45 - Precalculus with Calculus Previews

Question

Find the given trigonometric function value. Do not use a calculator. $$ \sin \left(-60^{\circ}\right) $$

Step-by-Step Solution

Answer

\( \sin(-60^{\circ}) = -\frac{\sqrt{3}}{2} \)

Step 1: Understand the Problem

We need to find the value of \( \sin(-60^{\circ}) \), which is a trigonometric function involving a negative angle.

Step 2: Use the Odd-Even Identity for Sine

The sine function is an odd function, which means \( \sin(-\theta) = -\sin(\theta) \). So, \( \sin(-60^{\circ}) = -\sin(60^{\circ}) \).

Step 3: Recall the Value of Sine for Positive Angle

For the angle \( 60^{\circ} \), the sine value is well-known: \( \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \).

Step 4: Apply the Odd Function Property

Thus, using Step 2, we can find \( \sin(-60^{\circ}) = -\sin(60^{\circ}) = -\frac{\sqrt{3}}{2} \).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, alongside cosine and tangent. It relates a given angle in a right-angled triangle to the ratio of the length of the triangle's opposite side over its hypotenuse. If you understand this definition, you can apply it to various geometrical situations involving angles. The sine function is very useful in many scientific fields, such as physics, engineering, and computer science. It helps model periodic phenomena, like sound waves and circular motion. These properties make it an essential tool in both theoretical and applied sciences.

• Defined as \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• Applicable in right-angled triangles and the unit circle
• Key in understanding periodicity and waveforms

Understanding the sine function is a crucial step in delving deeper into the world of trigonometry and related mathematical disciplines.

Angle Properties

Angles can be a little tricky because they have specific properties that determine how we calculate their trigonometric values. Firstly, remember that angles can be measured in different units, commonly degrees or radians. A full circle is 360 degrees or 2π radians, which is useful to know when dealing with trigonometric functions.

Also, every angle has both a positive and a negative equivalent. For instance, a positive angle rotates counterclockwise, while a negative one rotates clockwise. This property affects how functions like sine work, as we saw with \(\sin(-60^{\circ}) \). Angles are fundamental in trigonometric calculations and can impact the values of sine, cosine, and other functions.

• Can be in degrees or radians
• Positive angles rotate counterclockwise; negative, clockwise
• Affects trigonometric identities and function values

Mastering these properties will enhance your understanding of trigonometric calculations, making it easier to solve such exercises confidently.

Odd-Even Identity

The odd-even identity is an important concept that helps to simplify trigonometric problems, especially when dealing with negative angles. Trigonometric functions are categorized as either odd or even, which dictates how they behave with negative inputs. For an odd function, like sine, \(\sin(-\theta) = -\sin(\theta) \). This rule means if you know the sine of a positive angle, you can easily find the sine of its negative equivalent.

• Sine is an odd function: \(\sin(-\theta) = -\sin(\theta)\)
• Odd function identites flip the sign for negative angles
• Simplifies solving when angles are negative

Understanding the odd-even identity helps you in identifying and applying the correct simplifications to solve exercises without the need for a calculator. Once you grasp this concept, working through negative angles, like \(\sin(-60^{\circ}) \), becomes straightforward.