Radians and Reference angles

Question

How are radian measures more versatile than degree measures? How can reference angles help to quickly identify trig function values?

Answer 1

Radian is a "unit-less" unit of measure - its more like a percentage of a circle than anything else. When using it in an equation, we don't even include the unit in the math, it just disappears because it represents a portion of a circle and not a unit amount (it's almost like a placeholder).

Consequently, it's useful in performing calculations in which some equation is used. For instance, let's look at the arc length formula S = rθ where S is the length of an arc (portion of a circle) and θ is the angle the arc maps out. Like this:

For something to use, lets say this is a 60° angle or π/3 radians. Lets also say the radius is 1 cm long. So, if we used degrees we would say S = rθ = (1 cm)(60°) or S = 60 cm°. What the heck is a cm°?? Now lets use radians. S = rθ = (1cm)(π/3 radians) again radians is more like a placeholder so it just drops out of the equation so S = π/3 cm.

Reference angles are useful as you work with the unit circle. I suggest you wait until you've had the discussion of the unit circle to ask this question.

Answer 2

All I can tell you is that it depends on "formulas"; how they're written-out. Remember a radian is 180°/pi. A pi radian is 180°.

Answer 3

Start with the "unit circle", which is a convenient way to visualize the common trig functions.

Good article here:

https://www.mathsisfun.com/geometry/unit-circle.html

The use of radians vs. degrees depends on what you are doing. With some types of problems, the radian is convenient because--for small angles--the angle in radians is the same as the tangent of the angle.

Example:

0.1 radian = 5.7296^o tan(0.1) = 0.1003