AP Precalculus · Lesson 1

Radians, periodic behavior and trig values

Know Know it

An angle in radians is arc length over radius: θ=sr\theta = \frac{s}{r}. A full turn is 2π2\pi radians, so π\pi radians is 180 degrees. To change degrees to radians, multiply by π180\frac{\pi}{180}.

A periodic function repeats: f(x+p)=f(x)f(x + p) = f(x) for every x. The smallest such positive p is the period. Once you know one cycle you know every value, so to find a far-off value, first subtract whole periods.

Put an angle θ\theta in standard position. Its terminal ray meets the unit circle at (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta). For any point (x,y)(x, y) on the ray at distance r from the origin, sin⁡θ=yr\sin\theta = \frac{y}{r}, cos⁡θ=xr\cos\theta = \frac{x}{r} and tan⁡θ=yx\tan\theta = \frac{y}{x}, the slope of the ray.

Special values on the unit circle
θ\thetasin⁡θ\sin\thetacos⁡θ\cos\thetatan⁡θ\tan\theta
00001100
π6\frac{\pi}{6}12\frac{1}{2}32\frac{\sqrt{3}}{2}33\frac{\sqrt{3}}{3}
π4\frac{\pi}{4}22\frac{\sqrt{2}}{2}22\frac{\sqrt{2}}{2}11
π3\frac{\pi}{3}32\frac{\sqrt{3}}{2}12\frac{1}{2}3\sqrt{3}
π2\frac{\pi}{2}1100undefined

Trap Fix the sign after the reference angle

The reference angle gives the size of a trig value, not its sign. cos⁡2π3\cos\frac{2\pi}{3} has reference angle π3\frac{\pi}{3}, but the ray points into the second quadrant, where x is negative, so cos⁡2π3=−12\cos\frac{2\pi}{3} = -\frac{1}{2}, not 12\frac{1}{2}.

Worked example Easy

A wheel of radius 4 decimeters turns until a point on its rim has moved along an arc of 10 decimeters. Through what angle, in radians, did the wheel turn?

  1. 0.4
  2. 1.25
  3. 2.5Answer
  4. 40

How to solve it

  1. Use θ=sr\theta = \frac{s}{r} with s=10s = 10 and r=4r = 4.
  2. θ=104=2.5\theta = \frac{10}{4} = 2.5 radians.

Why each choice is right or wrong

  • A. Incorrect. 0.4 is the radius divided by the arc, which flips the ratio sr\frac{s}{r} upside down.
  • B. Incorrect. 1.25 divides the arc by the diameter, 8, but the angle is the arc over the radius.
  • C. Correct. The angle is sr=104=2.5\frac{s}{r} = \frac{10}{4} = 2.5 radians.
  • D. Incorrect. 40 multiplies the arc by the radius, but an angle in radians is the arc divided by the radius.

Explain Explain it

Task verbs: explain, calculate

To find a trig value of any angle, make four moves. Add or subtract whole turns of 2π2\pi to get an angle in [0,2π)[0, 2\pi). Name the quadrant. Find the reference angle. Read the size from the special values and attach the sign that x, y or the slope has in that quadrant.

Sine is positive in quadrants I and II, cosine in quadrants I and IV, and tangent in quadrants I and III, where x and y have the same sign.

Worked example Medium

What is the value of tan⁡11π6\tan\frac{11\pi}{6}?

  1. 33\frac{\sqrt{3}}{3}
  2. −3-\sqrt{3}
  3. −12-\frac{1}{2}
  4. −33-\frac{\sqrt{3}}{3}Answer

How to solve it

  1. 11π6\frac{11\pi}{6} is π6\frac{\pi}{6} short of 2π2\pi, so it is in quadrant IV with reference angle π6\frac{\pi}{6}.
  2. The point on the unit circle is (32,−12)\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right).
  3. tan⁡11π6=yx=−13=−33\tan\frac{11\pi}{6} = \frac{y}{x} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}.

Why each choice is right or wrong

  • A. Incorrect. 33\frac{\sqrt{3}}{3} has the right size, but 11π6\frac{11\pi}{6} ends in quadrant IV, where tangent is negative.
  • B. Incorrect. −3-\sqrt{3} divides x by y, which is the cotangent of the angle, not the tangent.
  • C. Incorrect. −12-\frac{1}{2} is sin⁡11π6\sin\frac{11\pi}{6}, the y-coordinate alone, not the ratio of y to x.
  • D. Correct. The reference angle is π6\frac{\pi}{6} and the angle is in quadrant IV, so the tangent is −1/23/2=−33\frac{-1/2}{\sqrt{3}/2} = -\frac{\sqrt{3}}{3}.

Explain Your turn

Explain why sin⁡(θ+2π)=sin⁡θ\sin(\theta + 2\pi) = \sin\theta for every angle θ\theta. Then use this fact to find sin⁡13π6\sin\frac{13\pi}{6}.

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Compare with a model answer

A full answer does this

  • Says that adding 2π2\pi is one full turn, so the ray ends at the same point of the unit circle, with the same y-coordinate
  • Finds sin⁡13π6=12\sin\frac{13\pi}{6} = \frac{1}{2} by subtracting 2π2\pi to reach π6\frac{\pi}{6}

Model answer

Adding 2π2\pi to an angle turns its ray one full turn, so the ray ends where it started and meets the unit circle at the same point. The sine is that point's y-coordinate, so it does not change. Since 13π6−2π=π6\frac{13\pi}{6} - 2\pi = \frac{\pi}{6}, we get sin⁡13π6=sin⁡π6=12\sin\frac{13\pi}{6} = \sin\frac{\pi}{6} = \frac{1}{2}.

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