Which trig functions are positive in each quadrant?

Cracking the Unit Circle: A Friendly Guide to Positive Trig Functions

Trigonometry can seem intimidating, right? But at its heart, it’s really about exploring the relationships between angles and the sides of triangles. And a super helpful tool for understanding all this is the unit circle. Think of it as a visual playground where we can see how trigonometric functions behave.

So, what’s the unit circle? Simply put, it’s a circle with a radius of 1, perfectly centered on a coordinate plane. It’s our map for navigating the world of sines, cosines, tangents, and their buddies: cosecant, secant, and cotangent. The unit circle is especially useful for figuring out when these trig functions are positive or negative, depending on where we are on the circle. Let’s break it down.

Quadrant by Quadrant

The x and y-axes chop up the coordinate plane into four sections, which we call quadrants. We label them I, II, III, and IV, going counter-clockwise, like the hands of a slightly rebellious clock. Each quadrant represents a slice of the circle, a specific range of angles:

• Quadrant I: This is where angles from 0° to 90° live (or 0 to π/2 radians, if you’re into that).
• Quadrant II: We’re talking angles from 90° to 180° here (π/2 to π radians).
• Quadrant III: This is the 180° to 270° zone (π to 3π/2 radians).
• Quadrant IV: Finally, we wrap things up with 270° to 360° (3π/2 to 2π radians).

ASTC: Your Trig Cheat Code

Now, how do we remember which trig functions are positive in each quadrant? Here’s a classic mnemonic that’s saved many a student: “All Students Take Calculus” (ASTC). Seriously, this little trick is gold.

• Quadrant I: All functions are positive. It’s a party in Quadrant I! Everyone’s invited, and everyone’s feeling good.
• Quadrant II: Students – Sine (and its reciprocal, cosecant) are the only ones in a positive mood here.
• Quadrant III: Take – Tangent (and cotangent, its reciprocal) are rocking the positive vibes in this quadrant.
• Quadrant IV: Calculus – Cosine (and its reciprocal, secant) are the positive players in the final quadrant.

The “Why” Behind the What

Okay, so we know what functions are positive where, but why does this pattern exist? It all boils down to the signs of the x and y-coordinates in each quadrant. Remember, on the unit circle, for any angle θ, we have a point (x, y). And guess what?

• cos(θ) is simply the x-coordinate.
• sin(θ) is the y-coordinate.
• tan(θ) is y divided by x (y/x).

Since cosecant, secant, and cotangent are just the flipped versions (reciprocals) of sine, cosine, and tangent, they follow the same sign rules.

• Quadrant I: Both x and y are positive, so everyone’s happy. All six trig functions are positive.
• Quadrant II: x is negative, but y is positive. So, sine (y) and cosecant (1/y) are positive, while the rest are negative.
• Quadrant III: Both x and y are negative. Tangent (y/x) and cotangent (x/y) are positive because a negative divided by a negative makes a positive!
• Quadrant IV: x is positive, but y is negative. That means cosine (x) and secant (1/x) are positive, while the others are negative.

Quick Reference

To make things even easier, here’s a handy table:

FunctionQuadrant IQuadrant IIQuadrant IIIQuadrant IVsin(θ)++–cos(θ)+–+tan(θ)+-+-csc(θ)++–sec(θ)+–+cot(θ)+-+-