Every trig computation this course does (how a ratio behaves near a point, how fast it's changing, how much area it sweeps out) is measured in radians, permanently, with no exceptions. The clean fact \frac{d}{dx}\sin x = \cos x is only true when x is in radians at all. Degrees don't disappear from the world, but they disappear from this course starting now. This is the last checkpoint to make sure the switch is automatic.
A radian isn't an arbitrary unit like a degree. 360 was chosen mostly because ancient Babylonian arithmetic liked the number. A radian is the angle "built into" the circle itself, measured with the circle's own radius as the ruler. That's why calculus prefers it. The rate-of-change formulas for sine and cosine come out clean in radians. That \pi is the same roughly-3.14 constant from a circle's circumference. In degrees they come out messy, with an extra leftover factor of \frac{\pi}{180} dangling on every one of them. A recipe reads cleaner in the units it was developed in, and this is the same idea.
A radian measures an angle by the length of the arc it cuts out of a circle, using the circle's own radius as the ruler. One radian is the angle for which the arc length exactly equals the radius. Since the full circumference of a circle is 2\pi times its radius, a full turn measures exactly 2\pi radians (\approx 6.28), the same full turn a degree scale calls 360°. That gives the conversion factor between the two systems. Multiplying a degree count by \pi/180 converts it to radians. Multiplying a radian count by 180/\pi converts it back. Which direction to multiply is easy to mix up. Multiplying by \pi/180 always shrinks the number, since 360 degrees becomes only about 6.28 radians, and the reverse conversion always grows it back. Checking that the result moved in the expected direction catches a flipped conversion before it propagates into every later step. This matters for more than bookkeeping: the clean fact \frac{d}{dx}\sin x = \cos x is only true when x is measured in radians.
Formula
Degree–Radian Conversion
Recalled from College Algebra · Radian measure — first taught there.
Both scales measure the same full rotation, just in different units: 360^\circ = 2\pi \text{ rad}. Divide both sides by 360 to find what one degree equals in radians: 1^\circ = \frac{2\pi}{360} \text{ rad} = \frac{\pi}{180} \text{ rad}. Multiplying by \theta_{\text{deg}} scales this up to any angle: \theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}. The reverse conversion is the reciprocal factor, \frac{180}{\pi}.
Formula
Exact Trig Values for Standard Angles
also known as: the unit circle table
Recalled from College Algebra · Reference angles and exact values — first taught there.
| \theta (deg) | \theta (rad) | \sin\theta | \cos\theta | \tan\theta |
|---|---|---|---|---|
| 0^\circ | 0 | 0 | 1 | 0 |
| 30^\circ | \pi/6 | 1/2 | \sqrt{3}/2 | \sqrt{3}/3 |
| 45^\circ | \pi/4 | \sqrt{2}/2 | \sqrt{2}/2 | 1 |
| 60^\circ | \pi/3 | \sqrt{3}/2 | 1/2 | \sqrt{3} |
| 90^\circ | \pi/2 | 1 | 0 | undefined |
| 120^\circ | 2\pi/3 | \sqrt{3}/2 | -1/2 | -\sqrt{3} |
| 135^\circ | 3\pi/4 | \sqrt{2}/2 | -\sqrt{2}/2 | -1 |
| 150^\circ | 5\pi/6 | 1/2 | -\sqrt{3}/2 | -\sqrt{3}/3 |
| 180^\circ | \pi | 0 | -1 | 0 |
| 210^\circ | 7\pi/6 | -1/2 | -\sqrt{3}/2 | \sqrt{3}/3 |
| 225^\circ | 5\pi/4 | -\sqrt{2}/2 | -\sqrt{2}/2 | 1 |
| 240^\circ | 4\pi/3 | -\sqrt{3}/2 | -1/2 | \sqrt{3} |
| 270^\circ | 3\pi/2 | -1 | 0 | undefined |
| 300^\circ | 5\pi/3 | -\sqrt{3}/2 | 1/2 | -\sqrt{3} |
| 315^\circ | 7\pi/4 | -\sqrt{2}/2 | \sqrt{2}/2 | -1 |
| 330^\circ | 11\pi/6 | -1/2 | \sqrt{3}/2 | -\sqrt{3}/3 |
| 360^\circ | 2\pi | 0 | 1 | 0 |
Every row reduces to a 0^\circ, 30^\circ, 45^\circ, 60^\circ, or 90^\circ reference angle with a quadrant sign attached. Nothing here is new information, just the special-triangle values and the sign table, tabulated once so you're not rederiving them mid-problem.
Formula
Reference Angle by Quadrant
Recalled from College Algebra · Reference angles and exact values — first taught there.
(replace 180° and 360° with \pi and 2\pi in radians).
Explore the special angles
Click around the circle and check the exact sine and cosine values at each of the sixteen special angles against the radian measures (\pi/6, \pi/4, \pi/3, and their multiples) from the technical block above.
Loading interactive…
Sine, measured the calculus way
Slide b — watch how many full waves fit in the same window.
Loading interactive…
Worked example
Converting and locating a reference angle
-
1
Find the radian measure of 210°, and its reference angle.
-
2
Multiply by \dfrac{\pi}{180}: 210° \times \dfrac{\pi}{180} = \dfrac{210\pi}{180} = \dfrac{7\pi}{6}.
-
3past π means quadrant III, where both sine and cosine are negative
\dfrac{7\pi}{6} is \pi + \dfrac{\pi}{6}: one full half-turn, plus a little more. That places it in the third quadrant, \dfrac{\pi}{6} past the negative x-axis.
past π means quadrant III, where both sine and cosine are negative
-
4
The reference angle is the acute angle to the nearest part of the x-axis: \dfrac{\pi}{6}.
Loading interactive…
Find the angle
Click the special angle whose sine equals 1/2.
Loading interactive…
Where this goes next
Chapter 1's key trigonometric behavior near zero
The single most important trig behavior this course studies is the ratio \frac{\sin x}{x} as x shrinks toward 0. It is a statement about radians specifically. That ratio settles toward exactly 1 in radians, but toward a much less useful number if x were measured in degrees. That one fact is where \frac{d}{dx}\sin x = \cos x ultimately comes from.
Check your understanding (counts toward your progress)
\cos\left(150^\circ\right) = \;?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Why
\cos\left(150^\circ\right) = -\sqrt{3}/2 — read the reference angle's magnitude off the special-triangle values, then apply this angle's own quadrant sign.
What is \cos\left(\dfrac{\pi}{3}\right)?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
\dfrac{\sqrt3}{2} is \sin(\pi/3), not \cos(\pi/3) -- the two special values for this angle are easy to swap.
\dfrac{\sqrt2}{2} belongs to \pi/4, a different special angle.
\cos(\pi/3) doesn't reach the maximum value 1 -- that only happens at \theta=0.
Why
On the unit circle, the point at \pi/3 is \left(\dfrac12,\dfrac{\sqrt3}{2}\right), so \cos(\pi/3)=\dfrac12.
Practice: Reference angles and radian sense (3 problems · doesn’t affect your progress)
What is the reference angle for \dfrac{5\pi}{4}?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
3\pi/4 would be the reference angle if this landed in quadrant II -- 5\pi/4 is in quadrant III instead.
The reference angle is always the acute angle measured to the NEAREST part of the x-axis, not the original angle itself.
\pi/2 is a quadrant boundary, not the acute angle from 5\pi/4 back to the nearest x-axis.
Why
\dfrac{5\pi}{4} = \pi + \dfrac{\pi}{4}, placing it in quadrant III exactly \dfrac{\pi}{4} past the negative x-axis -- so the reference angle is \dfrac{\pi}{4}.
5*pi/4 is pi + pi/4 -- one half-turn plus a little more, landing in quadrant III.
\dfrac{\pi}{4}
Find the reference angle for \dfrac{11\pi}{6}.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
\dfrac{11\pi}{6} is in quadrant IV, and 2\pi - \dfrac{11\pi}{6} = \dfrac{\pi}{6} is the acute angle back to the positive x-axis.
Correct answer
Also accepted:
Why
11pi/6 is in quadrant IV, just short of a full turn -- subtract it from 2pi to find the reference angle.
Which quadrant does \dfrac{4\pi}{3} fall in, compared to \dfrac{2\pi}{3}?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
This has the two swapped -- 4\\pi/3 is past \\pi, which is quadrant III, not II.
They don't share a quadrant -- 4\\pi/3 is past \\pi (quadrant III) while 2\\pi/3 is before it (quadrant II).
They don't share a quadrant -- 2\\pi/3 is between \\pi/2 and \\pi, which is quadrant II, not III.
Why
2\pi/3 sits between \pi/2 and \pi, which is quadrant II. 4\pi/3 sits between \pi and 3\pi/2, which is quadrant III.
Compare each angle to the quadrant boundaries pi/2, pi, and 3*pi/2.
4\pi/3 is in quadrant III; 2\pi/3 is in quadrant II
Quick review
Practice from earlier coursework, pulled in for this topic.
From College Algebra
Convert 315^\circ (three hundred fifteen degrees) to radians. Give an exact, fully reduced answer in terms of \pi.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
315 \times \frac{\pi}{180} = \frac{315\pi}{180}, which reduces to 7 \pi/4.
Correct answer
Also accepted:
Why
From College Algebra
Convert 330^\circ (three hundred thirty degrees) to radians. Give an exact, fully reduced answer in terms of \pi.
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
330 \times \frac{\pi}{180} = \frac{330\pi}{180}, which reduces to 11 \pi/6.
Correct answer
Also accepted:
Why
Quick review
Practice from earlier coursework, pulled in for this topic.
From College Algebra
What is the reference angle for \theta = 350^\circ (three hundred fifty degrees)?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
170^\circ would come from the Quadrant II formula, but 350^\circ is in Quadrant IV, which uses 360^\circ - \theta instead.
190^\circ isn't acute, and isn't produced by any of the reference-angle formulas for 350^\circ.
70^\circ isn't produced by any of the reference-angle formulas for 350^\circ either.
Why
350^\circ is in Quadrant IV, so the reference angle is 360^\circ - \theta = 360^\circ - 350^\circ = 10^\circ.
From College Algebra
Both coordinates of a unit-circle point in Quadrant III are negative. Which trig functions are positive there?
Right on try — credit. Right — the answer had already been shown, so no credit this time. Next try is worth Answer shown after three tries — this one no longer earns credit. You chose to see the answer — this one earns no credit.
\sin\theta = y, and y is negative throughout Quadrant III — sine can't be positive there.
\cos\theta = x, and x is negative throughout Quadrant III — cosine can't be positive there.
\sin\theta and \cos\theta are both negative in QIII (dividing a negative by a negative doesn't apply to them individually) — only the ratios y/x and x/y come out positive.
Why
In QIII, x < 0 and y < 0. So \sin\theta = y < 0 and \cos\theta = x < 0, but \tan\theta = y/x and \cot\theta = x/y divide a negative by a negative, giving a positive result.
Write out the signs of x and y in Quadrant III first, then check each function's definition (\sin=y, \cos=x, \tan=y/x, ...) against those signs.