Trigonometry Beyond Right Triangles · Lesson 1

Radians and the unit circle

A radian measures an angle by arc length: an angle of 1 radian cuts off an arc as long as the radius. A full turn is 2π2\pi radians, so 180∘=π180^\circ = \pi radians. To change degrees to radians, multiply by π180\frac{\pi}{180}; to change radians to degrees, multiply by 180π\frac{180}{\pi}. For example, 150∘=150⋅π180=5π6150^\circ = 150 \cdot \frac{\pi}{180} = \frac{5\pi}{6}.

Radians make arc length simple. On a circle of radius rr, a central angle θ\theta cuts off an arc of length s=rθs = r\theta, with θ\theta in radians. If the angle is given in degrees, convert it first, or use the fraction θ360\frac{\theta}{360} of the circumference.

The unit circle has radius 1 and its center at the origin. Turn an angle θ\theta counterclockwise from the positive x-axis; the point where its terminal side meets the circle is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta). Cosine is the x-coordinate, sine is the y-coordinate, and tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}.

Exact values to know
DegreesRadianscos⁡θ\cos\thetasin⁡θ\sin\theta
0∘0^\circ001100
30∘30^\circπ6\frac{\pi}{6}32\frac{\sqrt{3}}{2}12\frac{1}{2}
45∘45^\circπ4\frac{\pi}{4}22\frac{\sqrt{2}}{2}22\frac{\sqrt{2}}{2}
60∘60^\circπ3\frac{\pi}{3}12\frac{1}{2}32\frac{\sqrt{3}}{2}
90∘90^\circπ2\frac{\pi}{2}0011

Two triangles give the values at 30∘30^\circ, 45∘45^\circ and 60∘60^\circ; the 0∘0^\circ and 90∘90^\circ rows come from the points (1,0)(1, 0) and (0,1)(0, 1). A 45-45-90 triangle with hypotenuse 1 has legs of 22\frac{\sqrt{2}}{2}. A 30-60-90 triangle with hypotenuse 1 has a leg of 12\frac{1}{2} across from 30∘30^\circ and a leg of 32\frac{\sqrt{3}}{2} across from 60∘60^\circ.

Worked example Easy

What is the radian measure of an angle of 150∘150^\circ?

  1. 5π12\frac{5\pi}{12}
  2. 5π6\frac{5\pi}{6}Answer
  3. 5π3\frac{5\pi}{3}
  4. 27000π\frac{27000}{\pi}

How to solve it

  1. Degrees to radians: multiply by π180\frac{\pi}{180}.
  2. 150⋅π180=150π180150 \cdot \frac{\pi}{180} = \frac{150\pi}{180}.
  3. Divide the top and bottom by 30: 5π6\frac{5\pi}{6}.
  4. Check: 150∘150^\circ is a bit less than 180∘180^\circ, and 5π6\frac{5\pi}{6} is a bit less than π\pi.

Why each choice is right or wrong

  • A. Incorrect. 5π12\frac{5\pi}{12} is 150⋅π360150 \cdot \frac{\pi}{360}. It is 180∘180^\circ, not 360∘360^\circ, that equals π\pi, so dividing by 360 gives half the right answer.
  • B. Correct. 150⋅π180=5π6150 \cdot \frac{\pi}{180} = \frac{5\pi}{6}. As a check, 150∘150^\circ is just under 180∘180^\circ, and 5π6\frac{5\pi}{6} is just under π\pi.
  • C. Incorrect. 5π3\frac{5\pi}{3} is 150⋅π90150 \cdot \frac{\pi}{90}, which treats a right angle as π\pi. A right angle is π2\frac{\pi}{2}; π\pi goes with 180∘180^\circ.
  • D. Incorrect. 27000π\frac{27000}{\pi} is 150⋅180π150 \cdot \frac{180}{\pi}, the conversion run the wrong way. Going from degrees to radians, multiply by π180\frac{\pi}{180}.

Worked example Medium

A circle has a radius of 12 centimeters. What is the length, in centimeters, of the arc cut off by a central angle of 40∘40^\circ?

  1. 2π9\frac{2\pi}{9}
  2. 4π3\frac{4\pi}{3}
  3. 8π3\frac{8\pi}{3}Answer
  4. 480480

How to solve it

  1. Arc length s=rθs = r\theta needs θ\theta in radians.
  2. 40⋅π180=2π940 \cdot \frac{\pi}{180} = \frac{2\pi}{9}.
  3. s=12⋅2π9=24π9=8π3s = 12 \cdot \frac{2\pi}{9} = \frac{24\pi}{9} = \frac{8\pi}{3}.
  4. Check with a fraction of the circle: 40360⋅2π(12)=19⋅24π=8π3\frac{40}{360} \cdot 2\pi(12) = \frac{1}{9} \cdot 24\pi = \frac{8\pi}{3}.

Why each choice is right or wrong

  • A. Incorrect. 2π9\frac{2\pi}{9} is the angle in radians. That is one step short: multiply by the radius, 12⋅2π9=8π312 \cdot \frac{2\pi}{9} = \frac{8\pi}{3}.
  • B. Incorrect. 4π3\frac{4\pi}{3} comes from converting with π360\frac{\pi}{360}, which halves the angle. 40∘40^\circ is 40π180=2π9\frac{40\pi}{180} = \frac{2\pi}{9} radians, so the arc is 8π3\frac{8\pi}{3}.
  • C. Correct. In radians the angle is 2π9\frac{2\pi}{9}, and s=rθ=12⋅2π9=8π3s = r\theta = 12 \cdot \frac{2\pi}{9} = \frac{8\pi}{3}, about 8.4 centimeters.
  • D. Incorrect. 480480 is 12⋅4012 \cdot 40, which puts degrees into s=rθs = r\theta. The whole circumference is only 24π24\pi, about 75, so 480 can't be one arc.

Trap Converting the wrong way

Degrees to radians multiplies by π180\frac{\pi}{180}, so a radian answer usually has π\pi in it and a small number in front. If your radian answer comes out in the thousands, or your degree answer still has π\pi in it, you multiplied by the wrong fraction.

Trap Arc length with degrees

s=rθs = r\theta works only in radians. With r=12r = 12 and 40∘40^\circ, the product 12⋅40=48012 \cdot 40 = 480 is not an arc length: the whole circle is only 24π24\pi, about 75. Convert 40∘40^\circ to 2π9\frac{2\pi}{9} first.

Calculator When a calculator isn’t faster

Conversions and exact values are quicker by hand, and exact answers keep π\pi and radicals. When you do use a calculator for sine or cosine, check its angle mode first: degree/radian mode is the classic calculator trap. In the built-in Desmos calculator if you test online, the wrench (graph settings) switches between Radians and Degrees; on a handheld, look for DEG or RAD.

A quick mode test: sin⁡(30)\sin(30) gives 0.5 in degree mode. In radian mode it gives about −0.99-0.99, a sign that the mode is wrong for a degree question.