The unit circle (radius = 1) is the foundation of trigonometry. As an angle θ sweeps round the circle, the x-coordinate traces out cos θ and the y-coordinate traces out sin θ. The resulting waves are the basis of all engineering oscillations.
Unit circle — radius 1sin θ = y-coordinatecos θ = x-coordinatetan θ = sin θ / cos θ360° = full cycle
Drag the point or click anywhere on the circle
What the unit circle shows
For any angle θ, the point P on the unit circle has coordinates (cos θ, sin θ). This means:
cos θ = horizontal distance from centre to P
sin θ = vertical distance from centre to P
tan θ = sin θ / cos θ = vertical/horizontal
💡 The unit circle explains why trig values repeat every 360° — after a full rotation, P is back where it started.
Exact values — memorise these
sin 0°=0 sin 30°=½ sin 45°=√2/2 sin 60°=√3/2 sin 90°=1
cos 0°=1 cos 30°=√3/2 cos 45°=√2/2 cos 60°=½ cos 90°=0
tan 0°=0 tan 30°=1/√3 tan 45°=1 tan 60°=√3 tan 90°=∞
📌 Memory trick: sin values for 0°,30°,45°,60°,90° are √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 0.5, 0.71, 0.87, 1
📦
Quadrants & Sign Rules
The sign (+ or −) of sin, cos, and tan depends on which quadrant the angle is in. Knowing this lets you work out values for angles above 90° without a calculator.
Reference angle method
Every angle has a reference angle — the acute angle between the terminal side and the x-axis:
Q1 (0°–90°): ref = θ
Q2 (90°–180°): ref = 180° − θ
Q3 (180°–270°): ref = θ − 180°
Q4 (270°–360°): ref = 360° − θ
💡 Calculate trig of the reference angle, then apply the correct sign from the CAST rule.
Worked examples
sin 210°: Q3, ref=30°, sin negative → sin210° = −sin30° = −0.5
cos 135°: Q2, ref=45°, cos negative → cos135° = −cos45° = −0.707
tan 300°: Q4, ref=60°, tan negative → tan300° = −tan60° = −1.732
sin 330°: Q4, ref=30°, sin negative → sin330° = −sin30° = −0.5
∿
The Sine Wave — y = sin θ
The sine wave starts at 0, rises to +1 at 90°, returns to 0 at 180°, falls to −1 at 270°, and returns to 0 at 360°. It is the most fundamental waveform in engineering.
Drag the point on the circle · watch the sine wave trace out
Key properties of y = sin θ
Period = 360° (repeats every full turn)
Amplitude = 1 (peaks at +1 and −1)
Starts at 0, crosses zero at 0°, 180°, 360°
Maximum +1 at θ = 90°
Minimum −1 at θ = 270°
💡 Engineering form: y = A·sin(ωt + φ) A = amplitude, ω = angular frequency, φ = phase shift
Reading values from the graph
The graph shows the value of sin θ at a glance:
∿
The Cosine Wave — y = cos θ
The cosine wave starts at +1 at 0°, falls to 0 at 90°, reaches −1 at 180°, rises back to 0 at 270°, and returns to +1 at 360°. It is the sine wave shifted 90° to the left.
Drag the point on the circle · watch the cosine wave trace out
θ
sin θ
cos θ
tan θ
0°
0
+1
0
30°
+½
+√3/2
+1/√3
45°
+√2/2
+√2/2
+1
60°
+√3/2
+½
+√3
90°
+1
0
—
120°
+√3/2
−½
−√3
135°
+√2/2
−√2/2
−1
150°
+½
−√3/2
−1/√3
180°
0
−1
0
210°
−½
−√3/2
+1/√3
225°
−√2/2
−√2/2
+1
240°
−√3/2
−½
+√3
270°
−1
0
—
300°
−√3/2
+½
−√3
315°
−√2/2
+√2/2
−1
330°
−½
+√3/2
−1/√3
360°
0
+1
0
Key properties of y = cos θ
Period = 360°
Amplitude = 1
Starts at MAXIMUM (+1) at θ = 0°
Crosses zero at 90° and 270°
cos θ = sin(θ + 90°) — same shape, 90° phase shift
💡 Key identity: sin²θ + cos²θ = 1 (always true — comes from Pythagoras on the unit circle!)
Sine vs Cosine comparison
θ
sin θ
cos θ
0°
0
1
90°
1
0
180°
0
−1
270°
−1
0
360°
0
1
📌 When sin is at its peak (90°), cos is zero — and vice versa at 0°/360°. They are always 90° apart.
∿
The Tangent Wave — y = tan θ
Tan is very different from sin and cos. It has asymptotes at 90° and 270° where the value shoots off to ±infinity. Its period is only 180°.
Drag the point on the circle · asymptotes at 90° and 270°
Key properties of y = tan θ
Period = 180° (half that of sin and cos)
No maximum or minimum — goes to ±∞
Asymptotes (undefined) at 90°, 270°, 450°…
Crosses zero at 0°, 180°, 360°…
tan θ = sin θ / cos θ — undefined when cos θ = 0
⚠️ tan 90° is UNDEFINED — there is no answer. Your calculator gives an error or a very large number.
Why tan has asymptotes
tan θ = sin θ / cos θ. At 90°, cos 90° = 0. You cannot divide by zero — so tan 90° is undefined.
As θ → 90° from below: tan → +∞
As θ → 90° from above: tan → −∞
💡 In engineering, tan θ appears in the impedance triangle (tan φ = X/R), pipe slopes, and cutting tool angles. It is always used for angles well away from 90°.
θ
tan θ
0°
0
45°
1
89°
57.3 (very large)
90°
UNDEFINED
135°
−1
180°
0
📊
Comparing Sin, Cos and Tan
Sine (red), Cosine (navy) and Tan (purple, clipped) overlaid on one graph. See how they relate at every angle.
Key relationships
sin θ = cos(90° − θ) — complementary angles
cos θ = sin(90° + θ) — cos leads sin by 90°
tan θ = sin θ / cos θ
sin²θ + cos²θ = 1 — Pythagorean identity
sin and cos cross at 45° and 225° (value = ±√2/2 = ±0.707)
What to notice on the graph
🔴 Sin starts at 0, rises first
🔵 Cos starts at +1 (already at its peak)
🟣 Tan starts at 0 but rises much faster than sin
⚪ Where sin=0, tan=0 (at 0°, 180°, 360°)
⚪ Where cos=0, tan is undefined (at 90°, 270°)
⚪ Tan = 1 when sin = cos (at 45°, 225°)
⚙️
Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
⚡Electrical:Mains AC: V=325sin(100πt). 50 Hz, 230 V RMS.
🔊Acoustics:Sound: P=P₀sin(2πft). Read amplitude & phase from graph.
Two AC signals: V₁ = 10·sin(ωt) and V₂ = 10·sin(ωt + 60°). Find the resultant V₁ + V₂ at ωt = 30° and ωt = 90°.
[4]
At ωt=30°: V₁=10·sin30°=5.0V V₂=10·sin90°=10.0V Total=15.0 V At ωt=90°: V₁=10·sin90°=10.0V V₂=10·sin150°=5.0V Total=15.0 V
Q10
A cam follower has SHM profile y=(30/2)·(1−cos(πθ/90°)) mm for 0°≤θ≤90°. Find the displacement at θ=30° and θ=60°, and state where the follower is at half lift.
