Problem 16 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Sketch the angle. Then find its reference angle. \(150^{\circ}\)

Step-by-Step Solution

Answer

The reference angle for \(150^{\circ}\) is \(30^{\circ}\).

Step 1: Sketch the Angle

To sketch an angle of \(150^{\circ}\), start drawing from the positive x-axis and move in the counter-clockwise direction since it's positive angle. The final line should be in the second quadrant, halfway between the x- and y-axes.

Step 2: Identify the reference angle

The reference angle is the acute angle that the terminal line forms with the x-axis. The reference angle \(r\) in the second quadrant can be found using the formula \(r = 180^{\circ} - \text{angle} \). So here, \(r = 180^{\circ} - 150^{\circ} = 30^{\circ}\).

Step 3: Confirm that the reference angle is correct

The reference angle should be an acute angle (less than 90°). Ensure this is the case here, which it is, since the reference angle is 30 degrees, and it's calculated in the correct way.

Key Concepts

Angle Sketching

Sketching an angle is like drawing its picture on a coordinate plane. Start at the positive part of the x-axis, which is the horizontal line moving to the right. For a positive angle like \(150^{\circ}\), move counter-clockwise around the circle.

• Begin at 0° and swing your line towards the positive y-axis.
• Since \(150^{\circ}\) is more than \(90^{\circ}\), you move past the first quadrant.
• Stop in the second quadrant, where the angle is larger than a right angle but less than a straight line.

By successfully sketching like this, you visually represent the angle on the plane, making it easier to find reference angles later.
Just think of it like a clock's hand, where you start at 3 o'clock and move counter-clockwise to the spot between 11 o'clock and noon.

Acute Angle

An acute angle is always less than \(90^{\circ}\), like a sharp needle. When finding reference angles, it's important that they are acute, which means all reference angles are snug little angles smaller than a quarter of a full circle.
These angles are special because:

• They let us make sense of the actual angle's position.
• They help simplify calculations in trigonometry.

For our \(150^{\circ}\) angle, the reference angle was found to be \(30^{\circ}\), which is perfect because it's acute.
This acute reference angle allows us to reference the bigger angle and work with it more easily in calculations and understanding, especially when we're interested in sine, cosine, and other trigonometric functions.

Second Quadrant

The second quadrant is where angles between \(90^{\circ}\) and \(180^{\circ}\) live. It's like a neighborhood for angles that are more than a right angle but still less than a straight angle.
Here's what's important about the second quadrant:

• Angles here land on the negative x-axis, and their terminal side falls into this spot.
• Their reference angles are all calculated by subtracting the angle from \(180^{\circ}\).
• These angles help us work with negative x-values and positive y-values since they lie up top and to the left.

For a \(150^{\circ}\) angle, you'll find it nestled right in the second quadrant, making its reference angle \(30^{\circ}\). Understanding this placement will enhance your grasp of trigonometry and how angles relate to one another in a coordinate system.