Trigonometric Functions of General Angles and the Unit Circle SAT Notes

Right-triangle definitions extend to any angle by placing the angle in standard position and reading the coordinates of a point on its terminal side. The unit circle packages those definitions into exact coordinate values.

Trigonometric functions from a general point

Trigonometric functions from a point on a circleThe terminal side passes through P(−8, 15) on a circle of radius 17 in quadrant II.

Coordinate definitions on $x^2+y^2=r^2$

\[r=\sqrt{x^2+y^2},\qquad \sin\theta=\frac yr,\quad \cos\theta=\frac xr,\quad \tan\theta=\frac yx\ (x\ne0)\]

Worked example

Use a point that is not on the unit circle

The terminal side of $\theta$ passes through $$P(-8,15)$$ . Find sine, cosine, and tangent.

Radius
$r=\sqrt{(-8)^2+15^2}=\sqrt{289}=17$ .
Coordinate ratios
Use $$y/r$$ , $$x/r$$ , and $$y/x$$ .
Signs
Quadrant II has positive sine, negative cosine, and negative tangent.

\sin\theta=15/17

\cos\theta=-8/17

, and

\tan\theta=-15/8

The unit circle turns ratios into coordinates

Unit-circle point

\[x^2+y^2=1,\qquad P=(\cos\theta,\sin\theta)\]

Because $$r=1$$ , cosine is the $$x$$ -coordinate and sine is the $$y$$ -coordinate.

Master unit circleFamiliar angles from 0 degrees through 360 degrees with exact radian measures and coordinates.

Exact angle and coordinate data

0° = 0; (1, 0)
30° = π/6; (√3/2, 1/2)
45° = π/4; (√2/2, √2/2)
60° = π/3; (1/2, √3/2)
90° = π/2; (0, 1)
120° = 2π/3; (−1/2, √3/2)
135° = 3π/4; (−√2/2, √2/2)
150° = 5π/6; (−√3/2, 1/2)
180° = π; (−1, 0)
210° = 7π/6; (−√3/2, −1/2)
225° = 5π/4; (−√2/2, −√2/2)
240° = 4π/3; (−1/2, −√3/2)
270° = 3π/2; (0, −1)
300° = 5π/3; (1/2, −√3/2)
315° = 7π/4; (√2/2, −√2/2)
330° = 11π/6; (√3/2, −1/2)
360° = 2π; (1, 0)

Build exact values from special right triangles

30-60-90 exact-value triangleA right triangle with base √3, height 1, hypotenuse 2, and selected angle 30°.

45-45-90 exact-value triangleA right triangle with base 1, height 1, hypotenuse √2, and selected angle 45°.

First-quadrant exact values
Angle	Radians	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0^\circ$	$$0$$	$$0$$	$$1$$	$$0$$
$30^\circ$	$\pi/6$	$$1/2$$	$\sqrt3/2$	$\sqrt3/3$
$45^\circ$	$\pi/4$	$\sqrt2/2$	$\sqrt2/2$	$$1$$
$60^\circ$	$\pi/3$	$\sqrt3/2$	$$1/2$$	$\sqrt3$
$90^\circ$	$\pi/2$	$$1$$	$$0$$	undefined

Reference angles transfer exact values

Reference angle in quadrant IIThe 140 degree terminal side is 40 degrees from the negative x-axis.

Reference angle in quadrant IIIThe 220 degree terminal side is 40 degrees past the negative x-axis.

Reference angle in quadrant IVThe 310 degree terminal side is 50 degrees below the positive x-axis.

Evaluate with a reference angle

Locate the quadrant
Use the terminal side, not the size of the original rotation.
Find the acute reference angle
QII: $180^\circ-\theta$ ; QIII: $\theta-180^\circ$ ; QIV: $360^\circ-\theta$ .
Use the exact first-quadrant magnitude
Read the special-angle value for the reference angle.
Apply the sign
Determine the sign from the terminal side's quadrant.

Signs of sine, cosine, and tangent by quadrantText labels identify every sign pattern so the meaning does not depend on color.

Exact angle and coordinate data

45° = π/4; (√2/2, √2/2)
135° = 3π/4; (−√2/2, √2/2)
225° = 5π/4; (−√2/2, −√2/2)
315° = 7π/4; (√2/2, −√2/2)

Quadrant sign patterns from coordinate signs
Quadrant	Coordinate signs	Sine	Cosine	Tangent
I	$$x>0,y>0$$	Positive	Positive	Positive
II	$$x<0,y>0$$	Positive	Negative	Negative
III	$$x<0,y<0$$	Negative	Negative	Positive
IV	$$x>0,y<0$$	Negative	Positive	Negative

Coordinate-based sign logic

Sine

Sine follows $$y$$ : positive above the $$x$$ -axis, negative below it.

Cosine

Cosine follows $$x$$ : positive right of the $$y$$ -axis, negative left of it.

Tangent

Tangent is $$y/x$$ : positive when $$x,y$$ share a sign and negative when their signs differ.

Worked example

Evaluate an angle outside quadrant I

Find the exact values of $\sin 225^\circ$ , $\cos225^\circ$ , and $\tan225^\circ$ .

The reference angle is $225^\circ-180^\circ=45^\circ$ .
Quadrant III makes sine and cosine negative and tangent positive.
Use the $45^\circ$ magnitudes $\sqrt2/2,\sqrt2/2,1$ .

\sin225^\circ=-\sqrt2/2

\cos225^\circ=-\sqrt2/2

, and

\tan225^\circ=1

Axis angles require direct coordinate reasoning

Axis-angle values
Angle	Point	Sine	Cosine	Tangent
$$0$$ or $2\pi$	$$(1,0)$$	$$0$$	$$1$$	$$0$$
$\pi/2$	$$(0,1)$$	$$1$$	$$0$$	undefined
$\pi$	$$(-1,0)$$	$$0$$	$$-1$$	$$0$$
$3\pi/2$	$$(0,-1)$$	$$-1$$	$$0$$	undefined

Mini check

Use an exact reference value

What is $\cos300^\circ$ ?

$$1/2$$
$$-1/2$$
$\sqrt3/2$
$-\sqrt3/2$

Show answer and explanation

Answer: $$1/2$$

The reference angle is $60^\circ$ , and cosine is positive in quadrant IV.

Key takeaways

Unit-circle checklist

For a general point, first compute $r=\sqrt{x^2+y^2}$ .
On the unit circle, the terminal point is $(\cos\theta,\sin\theta)$ .
Use special triangles for exact magnitudes and quadrant coordinates for signs.
Reference angles are acute distances to the $$x$$ -axis.
At $$x=0$$ , tangent is undefined; at $$y=0$$ , tangent is zero.
Reduce coterminal angles before locating their terminal points.

Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.

Go to Practice Review Flashcards

Angle	Point	Sine	Cosine	Tangent
$\(0\)$ or $2\pi$	$\((1,0)\)$	$\(0\)$	$\(1\)$	$\(0\)$
$\pi/2$	$\((0,1)\)$	$\(1\)$	$\(0\)$	undefined
$\pi$	$\((-1,0)\)$	$\(0\)$	$\(-1\)$	$\(0\)$
$3\pi/2$	$\((0,-1)\)$	$\(-1\)$	$\(0\)$	undefined

Quadrant	Coordinate signs	Sine	Cosine	Tangent
I	$\(x>0,y>0\)$	Positive	Positive	Positive
II	$\(x<0,y>0\)$	Positive	Negative	Negative
III	$\(x<0,y<0\)$	Negative	Negative	Positive
IV	$\(x>0,y<0\)$	Negative	Positive	Negative