The Unit Circle and the Trigonometric Functions

Try this

Turn $P$ slowly from $0^{\circ}$ to $36 0^{\circ}$ . In which two quadrants does the red sine line $M P$ point up? In which two does the blue cosine line $O M$ point right?
Can the sine line or the cosine line ever be longer than $1$ ? What about the tangent line $A T$ ? Where is $P$ when the tangent line disappears?
Use the slider to push the angle past $36 0^{\circ}$ , then down to negative angles. For angles with the same terminal side, do the three readings change?
Switch on the graph of “ $y = sin x$ ”: each time $P$ turns a little, the point on the right moves forward a little. The height on the graph is the length of which segment on the circle?
The last reading is $sin^{2} θ + cos^{2} θ$ . Can you make it anything else?

Definition. Let the terminal side of any angle $α$ meet the unit circle at $P (x, y)$ . Then

sin α = y, cos α = x, tan α = \frac{y}{x} (x \neq = 0) .

The sine, cosine and tangent lines draw these three numbers as directed segments: the direction gives the sign and the length gives the size. Radian measure comes from the same picture: on the unit circle, the arc cut off by the angle $α$ has length equal to $α$ in radians, and one full turn is $2 π$ .

Predict first

In which quadrant does the terminal side of $θ = 100 0^{\circ}$ lie?

Quadrant IQuadrant IIQuadrant IIIQuadrant IV

Choose an answer first; then you can see the result.

Why: the reduction formulas are symmetries of the circle

The classroom rhyme “odd multiples switch, even ones stay; the sign follows the quadrant” does not need to be memorized. The unit circle has many symmetries: reflection in the $y$ -axis, reflection in the $x$ -axis, reflection in the origin, and rotation by $9 0^{\circ}$ about the origin. Each symmetry moves $P$ somewhere else, and however the coordinates change, the trig functions change the same way.

Start from α and visit four symmetric positions

Step 1 / 5

The starting point
Take an angle in quadrant I, $α = 3 5^{\circ}$ . Remember the two segments: the sine line points up, the cosine line points right.

$P (cos α, sin α)$
π − α: reflect in the y-axis
π + α: reflect in the origin
−α: reflect in the x-axis
π/2 + α: turn 90°, across becomes up

The same picture explains two more things:

$O P = 1$ , so the Pythagorean theorem gives $sin^{2} α + cos^{2} α = 1$ ; the similar triangles $△ O M P \sim △ O A T$ give $tan α = \frac{sin α}{cos α}$ .
The sine and cosine lines always stay within $[- 1, 1]$ , but the tangent line can be as long as you like: $sin x$ and $cos x$ are bounded, $tan x$ is not.

Solving trig inequalities on the unit circle

Solving $sin x ⩾ \frac{1}{2}$ means asking: which points of the unit circle have height at least $\frac{1}{2}$ ? Draw the horizontal line $y = \frac{1}{2}$ . It meets the circle at $3 0^{\circ}$ and $15 0^{\circ}$ , and the arc above it is the answer:

\frac{π}{6} + 2 k π ⩽ x ⩽ \frac{5 π}{6} + 2 k π (k \in Z) .

When several conditions must hold at once, stack the arcs (sectors) and take the part they share. For the domain of a complicated function, this is the safest method there is.

Drag

P

back and forth between

3 0^{\circ}

and

15 0^{\circ}

: the sine line

M P

never gets shorter than

\frac{1}{2}

Practice

Exercise 1Basic

Find $sin 75 0^{\circ}$ .

Exercise 2Basic

Find $tan (- 22 5^{\circ})$ .

Exercise 3Advanced

Given $sin α = \frac{3}{5}$ with $α$ in quadrant II, find $tan α$ .

Exercise 4Challenge

Given $θ \in (0, π)$ and $sin θ + cos θ = \frac{7}{13}$ , find $tan θ$ .

Try this

Why: the reduction formulas are symmetries of the circle

Start from α and visit four symmetric positions

Solving trig inequalities on the unit circle

Practice

Mistake Museum · Exhibits from this stop

Taking a square root without checking the quadrant

Treating tan 90° as a very large number