Angles In Standard Position & Coterminal Angles

by TextBrain Team 48 views

Alright, guys, let's dive into some angle action! We're going to tackle angles in standard position, figure out which quadrants they chill in, and find some coterminal buddies for each. Get your protractors and thinking caps ready!

I. Constructing Angles in Standard Position

So, what's this "standard position" thing all about? An angle in standard position has its vertex at the origin (that's the (0,0) point on your graph) and its initial side (the starting side) along the positive x-axis. From there, we rotate counterclockwise for positive angles and clockwise for negative angles. Let's break down each angle:

1) 190°

For this angle in standard position, start by drawing your x and y axes. Then, picture the positive x-axis as the starting point. Rotate counterclockwise. A 180° rotation would land you on the negative x-axis. Since we need to go to 190°, keep going another 10° past the negative x-axis. You'll end up in the third quadrant. Bam! Angle constructed.

2) 86°

Again, start with your axes and the positive x-axis. Rotate counterclockwise 86°. Since 90° gets you to the positive y-axis, you'll stop just before that. This angle lands squarely in the first quadrant. Easy peasy!

3) -142°

Now we're getting negative! This means we rotate clockwise from the positive x-axis. A -90° rotation puts you on the negative y-axis. Keep going past that another -52° (-142 - (-90) = -52). You will end up in the third quadrant. Remember, negative angles just mean we're rotating in the opposite direction.

4) π/3

Okay, let's switch gears to radians. Remember that π radians equals 180°. So, π/3 radians is (180°)/3 = 60°. That's a positive angle, so rotate counterclockwise 60° from the positive x-axis. You will end up in the first quadrant. This one's a common angle, so you'll see it a lot.

5) -π/3

Same as before, π/3 radians is 60°. But this time it's negative, so we rotate clockwise 60° from the positive x-axis. You will end up in the fourth quadrant.

6) -7π/4

This one looks a little trickier, but don't sweat it! First, let's convert to degrees: (-7π/4) * (180°/π) = -315°. That's a big negative angle. Rotate clockwise 315° from the positive x-axis. Another way to think about it is rotating a full circle (-360°) and then coming back 45° (-360° + 45° = -315°). You will end up in the first quadrant. Remember, a full rotation brings you right back where you started.

II. Quadrant Determination

Now, let's figure out which quadrant the terminal side (the ending side) of each angle lies in.

1) 380°

This is more than a full circle (360°). Subtract 360° to find a coterminal angle: 380° - 360° = 20°. So, 380° is coterminal with 20°. Since 20° is in the first quadrant, 380° is also in the first quadrant. Easy peasy lemon squeezy.

2) 230°

A 90° angle puts you on the positive y-axis, 180° on the negative x-axis, and 270° on the negative y-axis. 230° is between 180° and 270°, so it's in the third quadrant. No calculations needed here, just visualization!

3) -294°

Rotating clockwise 294°... that's a bit much to visualize directly. Let's find a positive coterminal angle. Add 360°: -294° + 360° = 66°. So, -294° is coterminal with 66°. 66° is in the first quadrant, so -294° is also in the first quadrant. The key is to find that positive coterminal angle to make it easier.

4) -π/3

We already discussed in part 1 that -π/3, which equals to -60 degrees ends in the fourth quadrant.

5) -7π/6

Let's convert this to degrees: (-7π/6) * (180°/π) = -210°. Add 360° to find a positive coterminal angle: -210° + 360° = 150°. 150° is between 90° and 180°, so it's in the second quadrant. Therefore, -7π/6 is also in the second quadrant.

6) π/2

π/2 radians is 90°. A 90° angle lies along the positive y-axis. While it doesn't lie within a quadrant, it's on the boundary between the first and second quadrants. Some people might say it's quadrantal, meaning it lies on an axis.

III. Finding Coterminal Angles

Coterminal angles are angles that share the same terminal side. They differ by a multiple of 360° (or 2π radians). To find coterminal angles, we just add or subtract 360° (or 2π) as many times as we want.

Let's find one positive and one negative coterminal angle for each:

1) 380°

  • Positive coterminal angle: 380° + 360° = 740°
  • Negative coterminal angle: 380° - 360° = 20°

Wait a sec! 20° is positive. Let's subtract another 360°: 20° - 360° = -340°.

So, 740° and -340° are coterminal with 380°.

2) 230°

  • Positive coterminal angle: 230° + 360° = 590°
  • Negative coterminal angle: 230° - 360° = -130°

3) -294°

  • Positive coterminal angle: -294° + 360° = 66°
  • Negative coterminal angle: -294° - 360° = -654°

4) -π/3

Let's work with radians. Remember 2π is a full circle.

  • Positive coterminal angle: -π/3 + 2π = -π/3 + 6π/3 = 5π/3
  • Negative coterminal angle: -π/3 - 2π = -π/3 - 6π/3 = -7π/3

5) -7π/6

  • Positive coterminal angle: -7π/6 + 2π = -7π/6 + 12π/6 = 5π/6
  • Negative coterminal angle: -7π/6 - 2π = -7π/6 - 12π/6 = -19π/6

6) π/2

  • Positive coterminal angle: π/2 + 2π = π/2 + 4π/2 = 5π/2
  • Negative coterminal angle: π/2 - 2π = π/2 - 4π/2 = -3π/2

Wrapping Up

There you have it! We've conquered angles in standard position, identified their quadrant locations, and found coterminal angles. Remember to visualize those rotations, and don't be afraid to convert between degrees and radians to make things easier. Keep practicing, and you'll be an angle master in no time!