Study Guide

Trigonometric Functions - The Unit Circle

The Unit Circle


The unit circle sounds so techno—like moon unit or parental unit—but it's not. Really. It's just a simple little circle with a radius of 1.



Here's what the unit circle looks like (we'll explain all those crazy-looking pieces in a minute):

What's the big deal?

It's a useful unit, that's what. With it, we can learn more about trig functions and better understand reference triangles.

It will also lead us to the radian, another angle measurement. We'll have to re-boot our brains to look at life in radians (rather than degrees).

Radians

Angles can be measured in degrees or in radians. Here's the lowdown on the radian phenom.

Let's draw a circle with radius r. Then mark off an arc on our circle with length r, like so.

The central angle that subtends our arc is equal to 1 radian.

Here's another way to look at this: a full circle has 360°, and a full circle has 2π radians.

360° = 2π radians

In other words, a half circle contains 180° or π radians.

Since they both equal half a circle, they must equal each other.

180° = π radians

Dividing both sides by 180° or dividing both sides by π radians yields a conversion factor equal to 1.

or

We can use this conversion factor to convert from degrees to radians, or from radians to degrees.

So there…not so bad, right? Try out these problems for size.

Sample Problem

Convert 30° to radians.

Multiply by the conversion factor. Which one? The one that has the unit we need on top. (Try topping that.) That way, the unit you're getting rid of will be in the denominator and will cancel.

We want the degrees to cancel out, so we'll multiply by . And since that's just a clever form of 1, we can multiply our angle by it without changing the angle.

There we go. It's the exact same angle, except it's in radian mode now.

Sample Problem

Convert 120° to radians.

Multiply by the conversion factor. Which one? The one that has the unit you need on top. That way, the unit you're getting rid of will be in the denominator and will cancel. (FYI— repeating stuff doesn't hurt, now does it?)

Sample Problem

Convert  radians to degrees.

This time, we want the radians to cancel out. We'll multiply by .

See ya later, π.

  • Quadrants


    The coordinate plane is split into four sections or quadrants, like so.

    Notice how 90° is right there at the positive y-axis. That's because 90° is exactly one-quarter of a full circle. We can also see that 180° sits right between Quadrant II and Quadrant III, and 270° separates Quadrant III and Quadrant IV.

    Acute angles (that is, smaller than 90°…and adorable) fall into Quadrant I. Angles larger than 90° fall into one of the other three quadrants. To simplify trigonometric expressions, we often rewrite non-acute angles as acute angles. To do this, we first need to learn all about reference angles.

    Reference Angles

    A reference angle is just the acute version of whatever angle we're looking at. It's the smallest angle that our angle makes with the x-axis. Let's use a ρ to represent our reference angles, which is just the common Greek letter "rho." As in, "Rho, rho, rho your boat."

    Since we're on a Greek fix, we'll use ɵ ("theta") to represent the actual angle.

    First, let's look at Quadrant I.

    No need for anything fancy in Quadrant I. Our reference angle is ρ = ɵ, because it's already an acute angle.

    On to Quadrant II. This one is a little trickier.

    In Quadrant II, ρ = 180° – ɵ or ρ = π – ɵ. In other words, to turn ɵ into ρ, we subtract ɵ from 180° (or from π radians if we're in radian mode).

    Now, Quadrant III.

    In this quadrant, ρ = ɵ – 180° or ρ = ɵ – π.

    Finally, Quadrant IV.

    In Quadrant IV, ρ = 360° – ɵ or ρ = 2π – ɵ.

    One little thing, though: our ɵ has some special requirements. It's gotta be within one full circle.

    0° ≤ ɵ ≤ 360° or 0 ≤ ɵ ≤ 2π

    If ɵ doesn't fall into this range, then we must add or subtract 360° or 2π, until we have a ɵ in the correct range.

    Now let's look at our six trig functions and see what their signs are for each quadrant.

    In Quadrant I, all six trig functions are positive.

    In Quadrant II, only sine and its reciprocal function cosecant are +. The other four trig functions are negative.

    In Quadrant III, only tangent and its reciprocal function cotangent are +.

    In Quadrant IV, only cosine and its reciprocal function secant are +.

    An easy way to remember this is ASTC (All, Sine, Tangent, Cosine), or All Students Take Calculus.

    Calm down; it's just a mnemonic device. And no, you don't have to take calculus—yet.

    Now, let's see what we can do with our newfound knowledge.

    Sample Problem

    Find the reference angle for 135°.

    First off, we need to figure out which quadrant we're in. Since 135° is more than 90° but less than 180°, our angle is in Quadrant II.

    That's 135° measured from the positive x-axis. We wanna turn that into an acute angle that's closer to the negative x-axis, because that angle will be smaller and easier to deal with.

    In Quadrant II, our formula is ρ = 180° – ɵ. Plug in ɵ = 135°.

    ρ = 180° – 135°
    ρ = 45°

    Sample Problem

    Find the reference angle for 330°.

    We've got an angle that's between 270° and a full 360°, so we're in Quadrant IV this time. Our formula is ρ = 360° – ɵ.

    ρ = 360° – 330°
    ρ = 30°

    Sample Problem

    Find the reference angle for 480°.

    Uh-oh...480° doesn't fall into our 0° ≤ ɵ ≤ 360° range.

    Since 480° is larger than 360°, we've gotta subtract 360° to get our ɵ.

    480° – 360° = 120°

    Much better: 120° falls into Quadrant II.

    In Quadrant II, ρ = 180° – ɵ. Plug and chug.

    ρ = 180° – 120°
    ρ = 60°

  • Special Angles Larger Than 90º

    Special Angles in Quadrant II

    Be forewarned: this involves lots of flipping (numbers, not burgers).

    Let's look at our 45°-45°-90°and 30°-60°-90° references triangles again. This time, let's flip them to Quadrant II:

    For the 45°-45°-90° triangle, ɵ = 135° and ρ = 45°.

    For this 30°-60°-90° triangle, ɵ = 120° and ρ = 60°.

    For this 30°-60°-90° triangle, ɵ = 150° and ρ = 30°.

    Special Angles in Quadrant III

    Now, flip your triangles over into Quadrant III.

    For the 45°-45°-90° triangle, ɵ = 225° and ρ = 45°.

    For this 30°-60°-90° triangle, ɵ = 210° and ρ = 30°.

    For this 30°-60°-90° triangle, ɵ = 240° and ρ = 60°.

    Special Angles in Quadrant IV

    Flip your triangles over into Quadrant IV for the grand finale.

    For the 45°-45°-90° triangle, ɵ = 315° and ρ = 45°.

    For this 30°-60°-90° triangle, ɵ = 300° and ρ = 60°.

    For this 30°-60°-90° triangle, ɵ = 330° and ρ = 30°.

  • Symmetry of Trigonometric Functions


    Think of poet William Blake.

    Then, think of that Blake poem The Tyger in all its stylistic, symmetrical, metered wonderfulness. After this dose of poetic license, you'll definitely be in the mood for trig functions and their fearful symmetry. 

    We can figure out the symmetry of the trig functions by comparing their values in Quadrant I and Quadrant IV.

    Start with a representative triangle in Quadrant I.

    In Quadrant I, That's just a rehashing of our basic trig ratios.

    Now, let's look at the same triangle flipped into Quadrant IV.

    In Quadrant IV, They're the same ratios as before, except we've got a negative y-value.

    Therefore sin(-ɵ) = -sin ɵ, making sine an odd function.

    And cos(-ɵ) = cos ɵ, making cosine an even function.

    Finally, tan(-ɵ) = -tan ɵ, making tangent an odd function, too.

    How 'bout the reciprocal functions? Since cosecant (csc) is the reciprocal of sine, cosecant is also an odd function.

    Since secant (sec) is the reciprocal of cosine, secant is also an even function.

    Last up, since cotangent (cot) is the reciprocal of tangent, cotangent is also an odd function.

    That's all she wrote.

  • Periodicity of Trig Functions


    Periodicity: it's oh-so-polysyllabic, but oh-so-simple. Just think of a unit circle with a moving line. That's it. Since our unit circle has a radius of 1, the sine function can be represented by a line.

    Or think of it this way: sine is always the opposite side over the hypotenuse, and the hypotenuse is always 1 in the unit circle (the hypotenuse is the radius). Anything over 1 is just itself (like 2/1 = 2), so when we're chilling inside the unit circle, the sine is just the length of the opposite side.

    Let's see what happens when we slide our angles around a bit.The sine line function is highlighted in red below.

    As our angle ɵ swings from 0 to (that's 0° to 90° for all you degree-lovers out there), sin ɵ swings from 0 to 1.

    Next, rotate ɵ from to π.

    This time, sin ɵ swings from 1 to 0.

    Similarly, as we swing from π to , sin ɵ swings from 0 to -1. (From flipping to swinging—now it's getting interesting, if not dizzying.)

    And as we swing from to 2π, sin ɵ swings from -1 to 0. Check it out:

    We can see from our unit circle that the amplitude (maximum value) of y = sin x is 1, and the period (time it takes for one full cycle) of y = sin x is 2π. After that, the whole process repeats again, which means the sine function is periodic. It repeats itself. It repeats itself.

    The cosine function is another periodic function.

    We can follow it by using the unit circle again. Since cosine is always the adjacent side over the hypotenuse, and since the hypotenuse is always 1 inside the unit circle, the cosine of any angle in the unit circle is just the length of the adjacent side.

    The cosine line function is highlighted in red.

    As ɵ swings from 0 to , cos ɵ swings from 1 to 0.

    As ɵ swings from to π, cos ɵ swings from 0 to -1.

    Similarly, as ɵ swings from π to , cos ɵ swings from -1 to 0.

    And as ɵ swings from to 2π, cos ɵ swings from 0 to 1.

    We can see from our unit circle that the amplitude (maximum value) of y = cos x is 1, and the period (distance it takes for one full cycle) of y = cos x is 2π.

    To sum things up, that means the sine or cosine of any angle will always have a value between -1 and 1, no matter how big or small the angle is.

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