Problem 18 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Sketch the angle. Then find its reference angle. \(-370^{\circ}\)

Step-by-Step Solution

Answer

The equivalent positive angle for -370 degrees is 350 degrees. The reference angle for this is 10 degrees.

Step 1: Converting the Negative Angle into Positive

If the angle is negative, add 360 degrees to the angle until the answer lies between 0 and 360 degrees.\n\( -370^{\circ} + 360^{\circ} = -10^{\circ} \)\nAgain we add 360 to -10 degrees.\n\-10^{\circ} + 360^{\circ} = 350^{\circ}\nSo, -370 degrees is the same as 350 degrees.

Step 2: Sketching the Angle

Place the angle in the appropriate quadrant on a graph and make a sketch. The positive equivalent angle of \(350^{\circ}\) will be in the IV quadrant.

Step 3: Finding the Reference Angle

The reference angle is the acute angle formed with the x-axis. For angles in the IV quadrant, the reference angle \(\Theta'\) is measured clockwise from the x-axis and is calculated by subtracting the angle from 360 degrees.\n\(\Theta' = 360^{\circ} - \Theta = 360^{\circ} - 350^{\circ} = 10^{\circ}\)\nSo, the reference angle for -370 degrees is 10 degrees.

Key Concepts

Negative Angle Conversion

In trigonometry, angles can be positive or negative. A negative angle typically means it is measured clockwise from the positive x-axis. To work with negative angles, it's often useful to convert them into positive angles by adding or subtracting full circles (360 degrees) until the angle falls into a standard range, usually between 0 and 360 degrees. Let's look at the example of \(-370^{\circ}\).

• First, you add 360 degrees to the negative angle: \(-370^{\circ} + 360^{\circ} = -10^{\circ}\)This doesn't yet give a positive angle, so we add 360 degrees again:
• \(-10^{\circ} + 360^{\circ} = 350^{\circ}\)
With these calculations, we've transformed a negative angle of \(-370^{\circ}\) into a positive angle of \(350^{\circ}\).

Converting negative angles can simplify where the terminal side of the angle lies on the circle.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. Understanding these quadrants helps in determining the sine, cosine, and other trigonometric function values of angles:

• **Quadrant I:** All trigonometric functions are positive here, consisting of angles from \(0^{\circ}\) to \(90^{\circ}\).
• **Quadrant II:** Sine is positive, others are negative. This quadrant covers angles from \(90^{\circ}\) to \(180^{\circ}\).
• **Quadrant III:** Tangent is positive, the others are negative, with angles between \(180^{\circ}\) and \(270^{\circ}\).
• **Quadrant IV:** Cosine is positive, others are negative. Angles here range from \(270^{\circ}\) to \(360^{\circ}\).

In our example, the positive angle equivalent to \(-370^{\circ}\) is \(350^{\circ}\), placing it in Quadrant IV. Recognizing the quadrant helps in calculating the reference angle and understanding its properties.

Angle Sketching

Sketching an angle involves placing its terminal side in the correct quadrant on a coordinate plane. The terminal side is determined by the angle's standard position, which initially has its vertex at the origin and its initial side along the positive x-axis. From there, the angle opens up according to its magnitude:- For \(350^{\circ}\), we imagine starting from the positive x-axis and rotating clockwise to the right into Quadrant IV.- The angle just falls short of a full circle by \(10^{\circ}\), as it is \(10^{\circ}\) shy of \(360^{\circ}\).By sketching this angle, it becomes easier to identify the reference angle. A reference angle is always measured between the terminal side and the x-axis, ensuring it's the smallest angle possible.In this sketch, the reference angle for \(350^{\circ}\) is calculated as:\[ \Theta' = 360^{\circ} - 350^{\circ} = 10^{\circ}\] making it evident that the reference angle fits neatly against the x-axis in Quadrant IV.Visualizing via sketches helps deepen understanding of angles and their placements on the plane.