Question

In Problems 39-56, use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \sin \left(-60^{\circ}\right) $$

Short answer

\( \text{sin}\big(-60^{\text{°}}\big) = -\frac{\bigsqrt{3}}{2} \)

Step-by-step solution

Identify the function

The given expression is \(\text{sin}\big(-60^{\text{°}}\big)\). The first step is to identify the trigonometric function involved, which in this case is sine.

Use the even-odd property

Recall that the sine function is an odd function, which means \(\text{sin}(-x) = -\text{sin}(x)\). Apply this property to the expression \(\text{sin}\big(-60^{\text{°}}\big)\): \[ \text{sin}\big(-60^{\text{°}}\big) = -\text{sin}\big(60^{\text{°}}\big) \]

Find the sine of 60°

Using the known value of \(\text{sin}\big(60^{\text{°}}\big)\), which is \[ \text{sin}\big(60^{\text{°}}\big) = \frac{\bigsqrt{3}}{2} \]

Substitute and simplify

Substitute \(\frac{\bigsqrt{3}}{2}\) into the expression from Step 2: \[ \text{sin}\big(-60^{\text{°}}\big) = -\frac{\bigsqrt{3}}{2} \]

Key concepts

Sine Function

The sine function is one of the main trigonometric functions. In a right triangle, the sine of an angle is calculated as the length of the opposite side divided by the length of the hypotenuse. This function is usually denoted as \(\text{sin}(x)\), where \(\text{x}\) is the angle. The sine function can also be defined for any real number using the unit circle. Here, the angle is measured from the positive x-axis. The sine of any angle gives the y-coordinate of the corresponding point on the unit circle. As angles increase, the sine function oscillates between -1 and 1, creating its characteristic wave-like shape.

Trigonometric Functions

Trigonometric functions are essential in mathematics, particularly in the study of triangles and modeling periodic phenomena. These functions include sine, cosine, and tangent, each related to specific ratios of a right triangle’s sides.
These functions can also be defined using a unit circle, which allows their application to any angle, not just those in right triangles. The unit circle approach makes trigonometric functions valuable for solving problems in calculus, physics, engineering, and more.
Understanding trigonometric functions includes knowing their properties, such as periodicity and symmetry. For instance, the sine function is periodic with a period of 360° or 2π radians. This means \(\text{sin}(x + 360°) = \text{sin}(x)\). Additionally, sine is an odd function, which means \(\text{sin}(-x) = -\text{sin}(x)\).

Angle in Degrees

Angles can be measured in degrees or radians, with 360 degrees being equivalent to 2π radians. Degrees are often used in geometry and trigonometry for their simplicity.
Within a right triangle, angles are crucial for calculating trigonometric ratios. For instance, a 60° angle in a right triangle relates directly to specific sine, cosine, and tangent values, based on the triangle's proportions.
Using degrees, the angle \(\text{-60°}\) may seem counterintuitive at first. However, it indicates clockwise rotation. Knowing how to handle negative angles and their trigonometric values is fundamental when applying even-odd properties, as with \(\text{sin}(-60°) = -\text{sin}(60°)\).

Exact Value

Exact values are used instead of approximations, particularly in mathematics, to provide more precise results. For certain angles, the trigonometric functions have well-known exact values.
In the context of trigonometry, memorizing the exact values for common angles like 0°, 30°, 45°, 60°, and 90° is essential.
For instance, the exact value of \(\text{sin}(60°)\) is \(\frac{\bigsqrt{3}}{2}\). Using these values makes complex calculations more manageable and reliable. Thus, knowing \(\text{sin}(-60°) = -\frac{\bigsqrt{3}}{2}\) helps solve problems accurately without needing a calculator.