Problem 11 - Precalculus Mathematics for Calculus

Question

Find the exact value of the trigonometric function. $$\text { sin } 150^{\circ}$$

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{2} \).

Step 1: Understand the Angle's Quadrant

The angle 150° is in the second quadrant because it is between 90° and 180°. In this quadrant, the sine function is positive.

Step 2: Determine the Reference Angle

A reference angle is the acute angle that the given angle makes with the x-axis. For 150°, the reference angle is found by subtracting it from 180°: 180° - 150° = 30°.

Step 3: Recall the Sine Value for the Reference Angle

The sine of 30° is a known value. From the special angles table or unit circle, the sine of 30° is \( \frac{1}{2} \).

Step 4: Apply the Sine Function Rule for the Quadrant

Since 150° is in the second quadrant where sine is positive, the sine of 150° will be the same as the sine of its reference angle, which is 30°. Therefore, \( \text{sin } 150^{\circ} = \text{sin } 30^{\circ} = \frac{1}{2} \).

Key Concepts

Sine Function

The sine function is one of the primary

• trigonometric functions
• used to describe the ratio of specific sides within right triangles.

It is particularly defined as the ratio of the length of the opposite side to the length of the hypotenuse.
In the realm of the unit circle, which is a circle with a radius of one, the sine of an angle equals the y-coordinate of the point on the circle.
This representation allows the sine function to be graphically depicted as a wave-like curve, exhibiting both positive and negative values as it moves through different quadrants.
In practical terms, for any given angle \(\theta\), the function can be denoted as \( \sin(\theta) \).
Understanding the sine function is crucial for solving various problems in trigonometry, physics, and engineering.

Reference Angle

The reference angle is the key to simplifying the understanding of trigonometric functions for any angle.
It is the acute angle—meaning it is always positive and lies between 0° and 90°—that a given angle makes with the x-axis.
To find the reference angle:

• In the first quadrant, the angle itself is the reference angle.
• In the second quadrant, subtract the angle from 180°.
• In the third quadrant, subtract 180° from the angle.
• In the fourth quadrant, subtract the angle from 360°.

The reference angle helps determine the trigonometric values in any quadrant by leveraging the properties of angles within the first quadrant, where we generally know exact trigonometric values.

Unit Circle

The unit circle is a powerful tool in trigonometry, providing a coherent way to remember and calculate the values of sine, cosine, and other trigonometric functions.
It is defined as a circle with a radius of one, centered at the origin of a coordinate plane.
The circle provides a visual representation, where each point

• on the circle corresponds to an angle, typically calculated in radians.
• gives sine and cosine as the y and x coordinates, respectively.

Using the unit circle, angles are easy to visualize and thus make it convenient to understand the periodic nature of the sine function.
This simplicity paves the way to solve trigonometric problems without merely relying on memorized values.

Quadrants in Trigonometry

The coordinate plane is split into four quadrants, each influencing the sign and characteristics of trigonometric functions, based on their positions.
The division is as follows:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

Understanding each quadrant is essential for predicting the sign of any trigonometric function at a specific angle.
For instance, with an angle of 150° residing in the second quadrant, we conclude that the sine is positive, aligning with the reference angle values of \(30^{\circ}\) which are typically used as benchmarks.