What are the quadrant angles?

Quadrantal Angles: Demystifying Those Axis-Hugging Angles

Angles are the bread and butter of trigonometry, and once you get the hang of them, you unlock a whole new level of mathematical understanding. Among the many types of angles you’ll encounter, quadrantal angles are pretty special. Why? Because they sit right on the axes of our coordinate plane, making their trig values super easy to figure out. Let’s dive in and see what makes them tick.

What Exactly Is a Quadrantal Angle?

Okay, so picture your standard x-y coordinate plane. A quadrantal angle is simply an angle that, when drawn in its “standard position” (vertex at the origin, initial side along the positive x-axis), ends up with its terminal side smack-dab on one of the axes. Think of it like this: the angle “hugs” either the x-axis or the y-axis.

These angles are always multiples of 90 degrees (or π/2 radians, if you’re a fan of radians). So, we’re talking about angles like 0°, 90°, 180°, 270°, and good old 360°. But here’s a neat trick: any angle that lands on the same axis as one of these is also a quadrantal angle. We call those “coterminal” angles. Imagine spinning around more than once – you can end up back on the same axis, right? For example, -90° is coterminal with 270°, and 450° is coterminal with 90°. They’re all part of the quadrantal angle club!

Quadrants and the Unit Circle: A Quick Refresher

Remember those four quadrants on the coordinate plane? They’re numbered I, II, III, and IV, going counter-clockwise. The unit circle, which is just a circle with a radius of 1 centered at the origin, helps us visualize angles and their trigonometric functions.

Each quadrant covers a specific range of angles:

• Quadrant I: A cozy spot for angles between 0° and 90°.
• Quadrant II: Where angles live between 90° and 180°.
• Quadrant III: Home to angles from 180° to 270°.
• Quadrant IV: The final stretch, from 270° to 360°.

Quadrantal angles are the VIPs that mark the boundaries between these quadrants. They’re the gatekeepers, if you will:

• 0° (or 360°): Chilling out on the positive x-axis.
• 90°: Standing tall on the positive y-axis.
• 180°: Hanging out on the negative x-axis.
• 270°: Relaxing on the negative y-axis.

Trig Functions and These Special Angles

Here’s where the magic happens. Remember that on the unit circle, any point on the circle can be described by the coordinates (cos θ, sin θ), where θ is the angle. This makes finding the sine and cosine of quadrantal angles a breeze! And once you know sine and cosine, you can figure out all the other trig functions.

Here’s a handy-dandy table summarizing the trig values for our quadrantal angles:

Angle (θ)sin θcos θtan θcsc θsec θcot θ0°010Undefined1Undefined90°10Undefined1Undefined0180°0-10Undefined-1Undefined270°-10Undefined-1Undefined0360°010Undefined1Undefined