Introduction: In this lesson, angles will be measured given information about the values of the fundamental trig functions. We will use calculator keystrokes as well as supporting illustrations drawn in a coordinate plane.
The Lesson: A diagram is shown below of a circle with radius 1 and an angle in standard position. P is the point where the angle intersects the circle and is 
.
Q is the point on the horizontal axis given by (1, 0). We ask two questions: - What is the measure of the reference angle?
- What is the measure of an angle A with this reference angle given that
?
To answer (i), since opp = 
and hyp = 1 we have 
. An angle with this sine is 60º. This is the reference angle. We can check this on a calculator. Using a TI-83 we would have: 
In mathematical notation this is 
. This is read “the inverse sine of 
” or “the angle with a sine of 
is 60º.”
To answer (ii) note that the reference angle is 60º. We add 60º to 360º getting A = 420º.
This type of diagram can be helpful in remembering which “special angles” have certain trig values. We will assume knowledge of the opp, adj and hyp sides of triangles involving 30º, 45º, and 60º. In the examples below, it will sometimes be helpful to remember the signs of trig functions in each quadrant. We use “ASTC” and remember this by “All Students Take Calculus” to recall that positive trig values are for All functions in quadrant one, Sine in quadrant two, Tangent in quadrant three, and Cosine in quadrant four.
Let's Practice:- An angle has a sine of
and is in quadrant two. What is the reference angle?
Using a calculator we have 
as a reference angle.
- An angle A has a tangent of
and 
. What is the measure of angle A?
Using a calculator we have 
which is the reference angle. To find angle A we add this to 180º getting A 
.
- Suppose cos(A) = 0.5. What is the reference angle?
Values of 
for sine and cosine come from a 
triangle. In this case with cosine, the adjacent side of the reference triangle is 1 and the hypotenuse is 2. Therefore the reference angle is 60º.
- Suppose the tangent of a quadrant four angle is -1. What is the reference angle and what would be a measurement of this quadrant four angle?
We can always get a reference angle by using the positive trig value, in this case 1. We have 
, or “an angle which has a tangent of 1 is 45º”. A quadrant four angle with this reference angle could be -45º or 315º.
- Suppose the cosine of a quadrant three angle is
. What is the reference angle and what could be the measurement of this quadrant three angle?
We can get the reference angle by using 
. In quadrant three, this angle could be 210º or -150º.
- An angle has a sine of -0.5. What is the reference angle? What could be the measurement of this angle?
The reference angle is 
. Since the sine is negative, this angle could be in either quadrants three or four. In quadrant three we could have 210º or -150º. In quadrant four we could have -30º or 330º.