Angle θ (Theta):
0°
0 rad
0° (0)
90° (π/2)
180° (π)
270° (3π/2)
360° (2π)
Special Exact Angles:
Quadrant I
Interactive Unit Circle
COORDINATES (cos θ, sin θ)
(
0.707
,
0.707
)
Touch & drag circle or rim
QUADRANT I
0° - 90°
ALL POSITIVE (+)
QUADRANT II
90° - 180°
SIN ONLY (+)
QUADRANT III
180° - 270°
TAN ONLY (+)
QUADRANT IV
270° - 360°
COS ONLY (+)
Triangle & Ratios
r = 1
EXACT FORM
Standard Angle (45°)
$\cos=\frac{\sqrt{2}}{2},\ \sin=\frac{\sqrt{2}}{2}$
SOH
⇄ CHO
$\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$
0.7071
Opp = 0.71
CHO
⇄ SOH
$\csc\theta = \frac{\text{Hyp}}{\text{Opp}}$
1.4142
Ratio: 1 / 0.71
CAH
⇄ SHA
$\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$
0.7071
Adj = 0.71
SHA
⇄ CAH
$\sec\theta = \frac{\text{Hyp}}{\text{Adj}}$
1.4142
Ratio: 1 / 0.71
TOA
⇄ CAO
$\tan\theta = \frac{\text{Opp}}{\text{Adj}}$
1.0000
Slope of radius
CAO
⇄ TOA
$\cot\theta = \frac{\text{Adj}}{\text{Opp}}$
1.0000
Ratio: 0.71 / 0.71
Identity:
$\sin^2\theta + \cos^2\theta = 1$
= 1.000
cos²θ = 0.500
sin²θ = 0.500
Wave Generator ($y = f(\theta)$)
As the unit circle point rotates, its vertical (sine) or horizontal (cosine) displacement unrolls over the period $[0, 2\pi]$.
$y = \sin(\theta)$
Swipe wave to scrub θ
Reciprocal Functions & Identities: CHO · SHA · CAO
Direct reciprocal flips of SOH · CAH · TOA ($\csc = \frac{\text{H}}{\text{O}}$, $\sec = \frac{\text{H}}{\text{A}}$, $\cot = \frac{\text{A}}{\text{O}}$)
Mission
Find an angle where $\sin\theta = 0.5$
Drag the point on the unit circle to the correct location.
Current: 0.707
CHO
Cosecant ($\csc\theta$)
Flip of SOH
$\csc\theta = \frac{\text{Hyp}}{\text{Opp}} = \frac{1}{\sin\theta}$
Ratio ($\frac{1}{\text{Opp}}$):
1 / 0.71
Value:
1.414
SHA
Secant ($\sec\theta$)
Flip of CAH
$\sec\theta = \frac{\text{Hyp}}{\text{Adj}} = \frac{1}{\cos\theta}$
Ratio ($\frac{1}{\text{Adj}}$):
1 / 0.71
Value:
1.414
CAO
Cotangent ($\cot\theta$)
Flip of TOA
$\cot\theta = \frac{\text{Adj}}{\text{Opp}} = \frac{1}{\tan\theta}$
Ratio ($\frac{\text{Adj}}{\text{Opp}}$):
0.71 / 0.71
Value:
1.000
45.0°
cos: 0.71
sin: 0.71