Problem 86 - Algebra and Trigonometry

Question

use reference angles to find the exact value of each expression. Do not use a calculator. $$ \sin \left(-\frac{35 \pi}{6}\right) $$

Step-by-Step Solution

Answer

The exact value of \(\sin \left(-\frac{35 \pi}{6}\right)\) is \(\frac{1}{2}(\sqrt{3})\).

Step 1: Convert the given radian measure to a related acute angle

To convert the given radian measure into a related acute angle, divide the negative angle -35π/6 by \(2\pi\) or \(\frac{35 \pi}{6} = - 35 \div 12 = -2.92\) radian.

Step 2: Determine the angle in the standard position

Since the sine function has a period of \(2\pi\), we need to bring the angle into an interval from 0 to \(2\pi\) or 0 to \(360^{\circ}\). The angle -2.92 radian can be transformed into the standard position by adding \(2\pi\) until it falls into this interval. Calculating this, we get \(2.92 + 2\pi = 2.92 + 6.28 = 0.36\) (or \(130^{\circ}\)). This means the position of the angle -35π/6 is the same as the angle at 0.36 radian or \(130^{\circ}\) in the standard position.

Step 3: Identify the reference angle

The reference angle is always the acute angle to the x-axis. For an angle in the second quadrant like \(130^{\circ}\), the reference angle can be found by subtracting it from \(180^{\circ}\). Calculating this, we get \(180^{\circ} - 130^{\circ} = 50^{\circ}\). Hence, the reference angle is \(50^{\circ}\).

Step 4: Use reference angles to find the value of the original angle

Because the original angle was in the second quadrant and sine is positive in the second quadrant, we take the positive value of the sine of the reference angle. We know from the unit circle that \(\sin(50^{\circ}) = \sin(\frac{5\pi}{6}) = \frac{1}{2}(\sqrt{3})\).

Key Concepts

Radian Measure

Radian measure is a way of expressing angles using the radius of a circle. In other words, it's an alternative to degrees, which most of us are more familiar with. A full circle is equal to around 6.28 radians, or more precisely, \(2\pi\) radians. This is because one complete revolution (360 degrees) is equal to the circle's circumference divided by its radius, which is \(2\pi\) times. To convert degrees to radians, you can use the formula:

• Radians = Degrees \(\times \frac{\pi}{180}\)

Conversely, to convert radians to degrees, you use:

• Degrees = Radians \(\times \frac{180}{\pi}\)

When dealing with radian measures, it is important to remember that negative radians indicate a clockwise rotation. For example, if you have an angle like \(-\frac{35 \pi}{6}\), the negative sign signifies that you are moving in the clockwise direction around the circle. Finding a related acute angle often involves converting a large or negative radian measure back into a positive measure that fits within a single circle, which is between \(0\) and \(2\pi\). This technique allows you to simplify complex problems by focusing on the essential angle within one circle revolution.

Sine Function

The sine function is a fundamental part of trigonometry. It relates to the right triangle, defining the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, the sine of an angle is simply the y-coordinate of the point on the unit circle that corresponds to that angle.Here are some key properties of the sine function:

• It is periodic with a period of \(2\pi\). This means its values repeat every \(2\pi\) radians.
• The sine of an angle is always a number between -1 and 1.
• In different quadrants of the circle, the sine function can be positive or negative. It's positive in the first and second quadrants and negative in the third and fourth.

For example, if you have a radian measure that places your angle in the second quadrant (between \(\frac{\pi}{2}\) and \(\pi\)), like our exercise with \(130^{\circ}\), the sine value will be positive. It is this behavior that allows us to simplify complex angles using reference angles, which will bring the focus to the sine value of a smaller, equivalent angle.

Unit Circle

The unit circle is a pivotal concept in trigonometry and a valuable tool for understanding angles and trigonometric functions. The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane.Key aspects of the unit circle include:

• The angle formed by the radius at any point on the circle corresponds to the angle measured from the positive x-axis.
• The coordinates of any point on the unit circle are \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle in radians.

A powerful aspect of the unit circle is that it allows you to find trigonometric function values at a glance once you understand how angles correspond to positions on the circle. For instance, with our specific problem, finding the angle of \(-\frac{35\pi}{6}\) eventually brings us to consider just the point it aligns with in the basic circle (hence the second quadrant's reference angle). When you know the basic angles and their sine and cosine values, you can apply transformations, such as shifts of \(2\pi\) or subtracting from \(\pi\) or \(2\pi\) to find equivalent angles quickly. This is how, in our example, we convert a complex angle back into a manageable related acute angle, allowing us to then find its sine using the unit circle.