Problem 15 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In Exercises 15-22, sketch the angle. Then find its reference angle. \(-100^{\circ}\)

Step-by-Step Solution

Answer

The sketch would be a line in the third quadrant with an anti-clockwise sweep of 260 degrees from the positive x-axis, and its reference angle is \(80^{\circ}\).

Step 1: Converting to a positive angle

The given angle is -100 degrees. The equivalent positive angle can be obtained by adding 360 degrees to the angle. Hence, the positive angle is \( -100^{\circ} + 360^{\circ} = 260^{\circ} \).

Step 2: Sketching the angle

To sketch 260 degree angle, start from the positive x-axis and measure the angle in the clockwise direction until 260 degrees is reached. The angle will lie in the third quadrant.

Step 3: Finding the reference angle

For angles in the third quadrant, the reference angle can be found by subtracting 180 degrees from the angle. Hence, the reference angle is \( 260^{\circ} - 180^{\circ} = 80^{\circ} \).

Key Concepts

Reference Angle

A reference angle is the smallest positive angle that forms between the terminal side of an angle and the x-axis. It's always measured as an acute angle, meaning it will be less than 90 degrees. Knowing the reference angle helps in simplifying trigonometric functions. It gives us equivalent angles with the same sine, cosine, or tangent values, typically found in the first quadrant.
To find the reference angle of an angle greater than 180 degrees:

• In Quadrant II, subtract the angle from 180 degrees.
• In Quadrant III, subtract 180 degrees from the angle.
• In Quadrant IV, subtract the angle from 360 degrees.

Understanding reference angles is key for converting angles and finding trigonometric values.

Quadrants

The coordinate plane is divided into four sections called quadrants, which are ordered counter-clockwise starting from the positive x-axis. Quadrants help in determining the sign of trigonometric functions, as each quadrant has specific positive and negative values for sine, cosine, and tangent.

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

In the context of angles, they help us visualize how far and in what direction to measure to reach a particular angle. For example, a 260-degree angle lies in Quadrant III.

Angle Conversion

Angle conversion involves finding equivalent angles within one complete rotation or converting angles between degrees and radians. For angles outside the standard range of 0 to 360 degrees, or 0 to 2π radians, it becomes essential to convert them into a more manageable form.
To convert a negative angle to a positive one, you can add 360 degrees repeatedly until you get a positive angle between 0 and 360 degrees. For example, to convert -100 degrees to a positive angle, you add 360 degrees to get 260 degrees.
Similarly, converting from degrees to radians involves multiplying by \( \frac{\pi}{180} \) and vice versa. Angle conversion is crucial in many real-world applications, especially where understanding the direction and magnitude of angles is needed.

Sketching Angles

Sketching angles is the process of graphically representing angles starting from the positive x-axis. It is an essential skill for visualizing and understanding angle placement in the coordinate system.

• Start sketching from the positive x-axis and rotate in the counter-clockwise direction for positive angles.
• For negative angles, rotate clockwise.
• To sketch an angle accurately, know its equivalent positive or reduced angle if it exceeds 360 degrees.
• The quadrant in which the terminal side lands helps visualize the angle’s trigonometric values.

For example, when sketching an angle of 260 degrees, begin at the positive x-axis and rotate clockwise through a complete circle and then further into Quadrant III. Knowing how to sketch angles accurately is fundamental in solving trigonometric problems and understanding geometrical properties.