Problem 3 - Algebra 2

Question

Find the exact value of each function. $$ \sin -240^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \(\sin(-240^\circ)\) is \(\frac{\sqrt{3}}{2}\).

Step 1: Determine the reference angle

To find the reference angle, first convert \(-240^\circ\) to a positive angle by adding \(360^\circ\). Calculating this: \(-240^\circ + 360^\circ = 120^\circ\). The reference angle for \(120^\circ\) is \(180^\circ - 120^\circ = 60^\circ\).

Step 2: Determine the quadrant

Since \(-240^\circ\) becomes \(120^\circ\) after conversion, and \(120^\circ\) lies in the second quadrant, this quadrant's sine function values are positive.

Step 3: Use the reference angle to find the sine value

The sine of \(60^\circ\) is known to be \(\sin 60^\circ = \frac{\sqrt{3}}{2}\). Since the angle \(120^\circ\) is in the second quadrant, the sine value remains positive.

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions. It is often abbreviated as \( \sin \) and helps in describing a circle's ratio related to angles. If you imagine a right triangle or even a circle, the sine of an angle inside is the ratio of the length of the opposite side to the hypotenuse. In simpler terms, it measures how "high" a point on the unit circle is above the x-axis.

• This function is periodic, repeating every \(360^\circ\) or \(2\pi\) radians.
• The range of the sine function is between \(-1\) and \(1\). So, it can never be more or less than these values.

When solving problems involving the sine function, it is crucial to take these properties into account. The sine function's value depends on the angle and the quadrant it is in. Depending on the quadrant, the sine can be positive or negative, which leads us to our next topic.

Reference Angle

Reference angles are helpful in simplifying the understanding and calculation of trigonometric functions. They are formed by taking any angle and measuring the smallest distance it makes with the horizontal (x-axis). Essentially, it's the "core" angle when you're interested in trigonometric values.For instance, when you encounter angles like \( -240^\circ \), the first step is to make it a manageable positive angle. Adding \(360^\circ\) transforms \(-240^\circ\) into \(120^\circ\). The reference angle helps in finding equivalent sine, cosine, and tangent values by comparing them with the special angles such as \(30^\circ, 45^\circ,\) and \(60^\circ\).

• Subtract the given angle from \(180^\circ\) if it's in the second quadrant.
• This angle guides in predicting the sine's positivity or negativity when observed in specific quadrants.

By understanding the reference angle, calculations become more straightforward and less prone to errors.

Quadrant Identification

Quadrants are the four sections of the coordinate plane. They help us determine the sign and value of trigonometric functions. Sine, cosine, and tangent functions have different signs depending on which quadrant the angle lies in. Each quadrant affects the sine value uniquely, especially in trigonometric problems.To identify the quadrant:

• First, always convert negative angles to positive by adding \(360^\circ\). For example, \( -240^\circ \) becomes \(120^\circ\).
• Second, observe where this new angle falls. Angles between \(90^\circ\) and \(180^\circ\) are located in the second quadrant, where sine is positive, and cosine is negative.

Identifying the correct quadrant ensures error-free results in solving problems, ensuring you apply the right sign for the sine function.

Angle Conversion

Angle conversion is sometimes needed when working with trigonometric functions. It involves transforming angles from negative to positive using a circle's total angle, \(360^\circ\), or vice-versa. This often leads to easier calculations and a better comprehension of the functions.

• To convert a negative angle like \(-240^\circ\), simply add \(360^\circ\): \( -240^\circ + 360^\circ = 120^\circ\).
• By converting angles, you can immediately see which quadrant the angle belongs to.
• This process does not change the sine or cosine values, but it allows you to work with comfortable and familiar positive angles.

When you convert angles, you're streamlining the calculation process and ensuring you apply correct trigonometric values accordingly. It's a crucial step to mastering trigonometry efficiently.