Lesson 19 · Trigonometric Functions

Trigonometric Functions

Master VB 2026's trig functions — Sin, Cos, Tan and their inverses, radian↔degree conversion, the unit circle, and real-world applications in graphics, navigation, physics, and wave generation.

« Lesson 18: Math Functions Lesson 20: Format Function »

Key Takeaway: All VB 2026 trig functions (Math.Sin, Math.Cos, Math.Tan) work in radians, not degrees. To convert: radians = degrees * Math.PI / 180 and degrees = radians * 180 / Math.PI. The inverse functions (Math.Asin, Math.Acos, Math.Atan, Math.Atan2) also return radians. Math.Atan2(y, x) is preferred over Math.Atan(y/x) because it handles all four quadrants and avoids division-by-zero when x=0.

Math.Sin(r)

Sine (radians in)

Sin(PI/2) = 1.0

Math.Cos(r)

Cosine (radians in)

Cos(0) = 1.0

Math.Tan(r)

Tangent (radians in)

Tan(PI/4) = 1.0

Math.Asin(x)

Inverse sine → radians

Asin(1) = PI/2

Math.Acos(x)

Inverse cosine → radians

Acos(1) = 0

Math.Atan(x)

Inverse tangent → radians

Atan(1) = PI/4

Math.Atan2(y,x)

Full-quadrant angle

Atan2(1,1) = PI/4

Math.Sinh(x)

Hyperbolic sine

Sinh(0) = 0

Math.Cosh(x)

Hyperbolic cosine

Cosh(0) = 1

Math.Tanh(x)

Hyperbolic tangent

Tanh(0) = 0

* Math.PI / 180

Degrees → radians

90° → PI/2

* 180 / Math.PI

Radians → degrees

PI/2 → 90°

19.1 Radians vs Degrees — The Critical Difference

A full circle is 360° or 2π radians. VB 2026's trig functions always expect radians. Forgetting to convert is the single most common trig bug in student code.

RadDeg.vb — Visual Basic 2026

' Conversion helpers -- define once, use everywhere
Private Function ToRad(degrees As Double) As Double
    Return degrees * Math.PI / 180
End Function

Private Function ToDeg(radians As Double) As Double
    Return radians * 180 / Math.PI
End Function

' Key radian values
ToRad(0)    '  0       radians  (0°)
ToRad(30)   '  PI/6    radians  (30°)
ToRad(45)   '  PI/4    radians  (45°)
ToRad(60)   '  PI/3    radians  (60°)
ToRad(90)   '  PI/2    radians  (90°)
ToRad(180)  '  PI      radians  (180°)
ToRad(360)  '  2*PI    radians  (360°)

' Common trig values (exact)
Math.Sin(ToRad(0))    ' 0.0
Math.Sin(ToRad(30))   ' 0.5
Math.Sin(ToRad(45))   ' 0.7071... (√2/2)
Math.Sin(ToRad(90))   ' 1.0
Math.Cos(ToRad(0))    ' 1.0
Math.Cos(ToRad(60))   ' 0.5
Math.Cos(ToRad(90))   ' ≈ 0  (floating-point near-zero)
Math.Tan(ToRad(45))   ' 1.0

' WRONG -- passing degrees directly without conversion
Math.Sin(90)   ' 0.8939...  NOT 1.0  (treating 90 as 90 radians!)

The #1 Trig Bug

Passing degrees directly to Math.Sin, Math.Cos, or Math.Tan produces wrong results silently — no exception is thrown. Always convert with degrees * Math.PI / 180 before calling a trig function, or define a ToRad helper at the top of your form and use it everywhere.

19.2 Sin, Cos, and Tan

Angle	Math.Sin	Math.Cos	Math.Tan	Notes
0°	0	1	0	Start of unit circle
30°	0.5	0.866	0.577	sin = 1/2
45°	0.707	0.707	1	sin = cos = √2/2
60°	0.866	0.5	1.732	cos = 1/2
90°	1	≈0	±∞	Tan undefined at 90°
180°	≈0	-1	≈0	Half circle
270°	-1	≈0	±∞	Tan undefined at 270°
360°	≈0	1	≈0	Full circle = 0°

SinCosTan.vb — Visual Basic 2026

' Right-triangle side calculator
' Given hypotenuse and angle (degrees), find opposite and adjacent sides
Private Sub CalcTriangle(hyp As Double, angleDeg As Double)
    Dim rad      = ToRad(angleDeg)
    Dim opposite = Math.Round(hyp * Math.Sin(rad), 4)
    Dim adjacent = Math.Round(hyp * Math.Cos(rad), 4)
    Dim tangent  = Math.Round(Math.Tan(rad), 4)

    lblOpposite.Text = opposite.ToString()
    lblAdjacent.Text = adjacent.ToString()
    lblTangent.Text  = tangent.ToString()
End Sub

' Pythagorean identity: sin²θ + cos²θ = 1 (always true)
For deg As Integer = 0 To 360 Step 15
    Dim r = ToRad(deg)
    Dim identity = Math.Pow(Math.Sin(r), 2) + Math.Pow(Math.Cos(r), 2)
    ' identity ≈ 1.0 for all angles
Next

' Plotting a sine wave to a PictureBox
' x goes 0 to 360 degrees; y = A * sin(f * rad) + offset
Private Sub DrawSineWave(g As Graphics, w As Integer, h As Integer)
    Dim cx = w \ 2, cy = h \ 2      ' centre of drawing area
    Dim amp = cy - 10               ' amplitude (pixels)
    Dim pts(360) As PointF
    For i = 0 To 360
        Dim x = CSng(i / 360.0 * w)
        Dim y = CSng(cy - amp * Math.Sin(ToRad(i)))
        pts(i) = New PointF(x, y)
    Next
    g.DrawLines(Pens.Blue, pts)
End Sub

Try It — Simulation 19.1: Unit Circle Explorer

Drag the angle slider or type a degree value. The unit circle updates in real time showing sin, cos, and tan values, with the point on the circle and the right-triangle sides.

Unit Circle Explorer

Angle (degrees):

degrees

19.3 Inverse Trig Functions — Asin, Acos, Atan, Atan2

The inverse trig functions answer "what angle has this sin/cos/tan value?" They all return radians. Math.Atan2(y, x) is especially important in graphics programming because it determines the angle of a vector (from origin to point) and correctly handles all four quadrants.

InverseTrig.vb — Visual Basic 2026

' Math.Asin -- inverse sine, returns [-PI/2, PI/2]
ToDeg(Math.Asin(0))    '  0°
ToDeg(Math.Asin(0.5))  '  30°
ToDeg(Math.Asin(1))    '  90°
ToDeg(Math.Asin(-1))   ' -90°
Math.Asin(2)            ' Double.NaN  (domain: -1 to 1 only)

' Math.Acos -- inverse cosine, returns [0, PI]
ToDeg(Math.Acos(1))    '   0°
ToDeg(Math.Acos(0.5))  '  60°
ToDeg(Math.Acos(0))    '  90°
ToDeg(Math.Acos(-1))   ' 180°

' Math.Atan -- inverse tangent, returns (-PI/2, PI/2)
' WARNING: does not distinguish quadrants!
ToDeg(Math.Atan(1))    '  45°  (same as Atan(-1,-1) in quadrant 3!)
ToDeg(Math.Atan(-1))   ' -45°

' Math.Atan2(y, x) -- PREFERRED, full (-PI, PI] range, all quadrants
ToDeg(Math.Atan2(1, 1))    '  45°  (Q1: x>0, y>0)
ToDeg(Math.Atan2(1, -1))   ' 135°  (Q2: x<0, y>0)
ToDeg(Math.Atan2(-1, -1))  ' -135° (Q3: x<0, y<0)
ToDeg(Math.Atan2(-1, 1))   '  -45° (Q4: x>0, y<0)
ToDeg(Math.Atan2(1, 0))    '  90°  (x=0: no div-by-zero!)

' Practical: angle between two screen points
Private Function AngleBetween(x1 As Double, y1 As Double,
                               x2 As Double, y2 As Double) As Double
    Return ToDeg(Math.Atan2(y2 - y1, x2 - x1))
End Function

' Find missing angle in right triangle given two sides
Private Function AngleFromSides(opposite As Double, hypotenuse As Double) As Double
    Return ToDeg(Math.Asin(opposite / hypotenuse))
End Function

AngleFromSides(3, 5)   ' 36.87°  (3-4-5 right triangle)

19.4 Wave Generation

Sine and cosine are periodic functions — they repeat every 2π radians. By adjusting amplitude (height), frequency (how fast it repeats), and phase (where it starts), you can model any periodic phenomenon: sound waves, oscillating springs, rotating objects, and game animations.

Waves.vb — Visual Basic 2026

' General wave formula: y = A * sin(2π * f * t + φ)
' A = amplitude, f = frequency (Hz), t = time (sec), φ = phase (radians)

Private Function SineWave(t As Double, amplitude As Double,
                            frequency As Double,
                            Optional phase As Double = 0) As Double
    Return amplitude * Math.Sin(2 * Math.PI * frequency * t + phase)
End Function

' Lissajous figure: x = A*cos(a*t + δ), y = B*sin(b*t)
' When a:b = 1:2, produces a figure-8
Private Sub DrawLissajous(g As Graphics, cx As Integer, cy As Integer,
                             a As Integer, b As Integer)
    Dim pts(629) As PointF
    For i = 0 To 629
        Dim t = i * 0.01       ' t from 0 to 2π (approx)
        Dim x = CSng(cx + 90 * Math.Cos(a * t + Math.PI / 2))
        Dim y = CSng(cy + 90 * Math.Sin(b * t))
        pts(i) = New PointF(x, y)
    Next
    g.DrawLines(Pens.Blue, pts)
End Sub

' Circular motion: object orbiting a center point
' Used in game engines, clock hands, radar sweeps
Private Sub UpdateOrbit(t As Double, cx As Integer, cy As Integer,
                          radius As Double, speed As Double)
    Dim angle = t * speed    ' radians -- advances with time
    Dim x = cx + CInt(radius * Math.Cos(angle))
    Dim y = cy + CInt(radius * Math.Sin(angle))
    picObject.Location = New Point(x, y)
End Sub

Try It — Simulation 19.2: Wave Generator

Adjust amplitude, frequency, phase, and wave type to see how each parameter affects the curve. Overlay Sin + Cos to see phase relationships.

Wave Generator

Amplitude: 80

Frequency: 1

Phase (deg): 0

Wave Type:

19.5 Navigation — Bearing and Distance

Navigation.vb — Visual Basic 2026

' Haversine formula: great-circle distance between two GPS coordinates
' Returns distance in kilometres
Private Function Haversine(lat1 As Double, lon1 As Double,
                             lat2 As Double, lon2 As Double) As Double
    Const R = 6371.0   ' Earth radius in km
    Dim dLat = ToRad(lat2 - lat1)
    Dim dLon = ToRad(lon2 - lon1)
    Dim a = Math.Pow(Math.Sin(dLat / 2), 2) +
             Math.Cos(ToRad(lat1)) * Math.Cos(ToRad(lat2)) *
             Math.Pow(Math.Sin(dLon / 2), 2)
    Dim c = 2 * Math.Atan2(Math.Sqrt(a), Math.Sqrt(1 - a))
    Return Math.Round(R * c, 2)
End Function

' Compass bearing from point 1 to point 2 (degrees from North, 0-360)
Private Function Bearing(lat1 As Double, lon1 As Double,
                           lat2 As Double, lon2 As Double) As Double
    Dim r1 = ToRad(lat1), r2 = ToRad(lat2)
    Dim dL = ToRad(lon2 - lon1)
    Dim y  = Math.Sin(dL) * Math.Cos(r2)
    Dim x  = Math.Cos(r1) * Math.Sin(r2) - Math.Sin(r1) * Math.Cos(r2) * Math.Cos(dL)
    Return (ToDeg(Math.Atan2(y, x)) + 360) Mod 360
End Function

' Example: Kuala Lumpur to Singapore
Dim km = Haversine(3.1390, 101.6869, 1.3521, 103.8198)  ' ≈ 319 km
Dim bearing = Bearing(3.1390, 101.6869, 1.3521, 103.8198) ' ≈ 156° (SSE)

Try It — Simulation 19.3: Triangle Solver

Solve a right triangle using trig functions. Enter any two known values (angle + side, or two sides) and compute all missing sides, angles, area, and perimeter. Visual diagram updates in real time.

Right Triangle Solver

Known:

Hypotenuse:

Angle A (deg):

19.6 Practical: Analogue Clock & Rotation

AnalogClock.vb — Visual Basic 2026

' Analogue clock hand positions
' Clock hands rotate clockwise; 12 o'clock is -90° (top = -PI/2 radians)

Private Function HandEndPoint(cx As Integer, cy As Integer,
                               length As Integer, angleDeg As Double) As Point
    Dim rad = ToRad(angleDeg - 90)   ' -90 shifts 0° from right to top
    Return New Point(cx + CInt(length * Math.Cos(rad)),
                       cy + CInt(length * Math.Sin(rad)))
End Function

Private Sub DrawClock(g As Graphics, now As DateTime)
    Dim cx = 100, cy = 100   ' centre
    ' Hour hand: 360° / 12h = 30° per hour, +0.5° per minute
    Dim hourAngle = now.Hour * 30 + now.Minute * 0.5
    ' Minute hand: 360° / 60min = 6° per minute
    Dim minAngle  = now.Minute * 6
    ' Second hand: 6° per second
    Dim secAngle  = now.Second * 6

    g.DrawLine(Pens.Black, New Point(cx, cy), HandEndPoint(cx, cy, 55, hourAngle))
    g.DrawLine(Pens.DarkBlue, New Point(cx, cy), HandEndPoint(cx, cy, 75, minAngle))
    g.DrawLine(Pens.Red,   New Point(cx, cy), HandEndPoint(cx, cy, 80, secAngle))
End Sub

' Rotate a point around an origin (2D rotation matrix)
' x' = x*cos(θ) - y*sin(θ)
' y' = x*sin(θ) + y*cos(θ)
Private Function RotatePoint(px As Double, py As Double,
                               ox As Double, oy As Double,
                               angleDeg As Double) As PointF
    Dim rad = ToRad(angleDeg)
    Dim dx = px - ox, dy = py - oy
    Return New PointF(CSng(ox + dx * Math.Cos(rad) - dy * Math.Sin(rad)),
                       CSng(oy + dx * Math.Sin(rad) + dy * Math.Cos(rad)))
End Function

Try It — Simulation 19.4: Live Analogue Clock

A live analogue clock drawn entirely with trig functions. Each hand position is computed using Math.Sin and Math.Cos with the degree-to-radian conversion. Shows the VB formula for each hand.

Analogue Clock — Powered by Math.Sin / Math.Cos

Try It — Simulation 19.5: Trig Function Explorer

Enter any angle (degrees or radians) and see all six trig values: sin, cos, tan, asin, acos, atan. Includes the inverse calculation and the VB code for each call.

Trig Function Explorer

Angle value:

Units:

19.7 GitHub Copilot — Trig Patterns

GitHub Copilot — Satellite Dish Pointing

You typed:

' Given a satellite position (azimuth and elevation in degrees), calculate the dish motor angles needed to point at it. Use Math.Sin, Math.Cos, Math.Atan2.'

' Satellite dish pointing calculator -- azimuth/elevation to motor angles.
Private Sub PointDish(azimuthDeg As Double, elevationDeg As Double)
    Dim az = ToRad(azimuthDeg)
    Dim el = ToRad(elevationDeg)

    ' Convert spherical (az, el) to Cartesian unit vector
    Dim x = Math.Cos(el) * Math.Sin(az)
    Dim y = Math.Cos(el) * Math.Cos(az)
    Dim z = Math.Sin(el)

    ' Back to polar for motor control
    Dim motorAz = ToDeg(Math.Atan2(x, y))
    Dim motorEl = ToDeg(Math.Asin(z))
    ' Normalise azimuth to 0–360
    If motorAz < 0 Then motorAz += 360

    lblMotorAz.Text = $"Motor Azimuth: {motorAz:F2}°"
    lblMotorEl.Text = $"Motor Elevation: {motorEl:F2}°"
End Sub

GitHub Copilot — Projectile Motion

You typed:

' Simulate projectile motion: given initial speed and launch angle, calculate range, max height, and flight time. Use Math.Sin, Math.Cos, Math.Pow. Ignore air resistance.'

' Projectile motion calculator (no air resistance).
Private Sub CalculateProjectile(speed As Double, angleDeg As Double)
    Const g = 9.81
    Dim rad = ToRad(angleDeg)

    ' Horizontal and vertical velocity components
    Dim vx = speed * Math.Cos(rad)          ' horizontal (constant)
    Dim vy = speed * Math.Sin(rad)          ' vertical (decelerates)

    ' Flight time: T = 2*vy / g
    Dim T = Math.Round(2 * vy / g, 3)

    ' Maximum height: H = vy² / (2g)
    Dim H = Math.Round(Math.Pow(vy, 2) / (2 * g), 3)

    ' Horizontal range: R = vx * T  (or v²*sin(2θ)/g)
    Dim R = Math.Round(Math.Pow(speed, 2) * Math.Sin(2 * rad) / g, 3)

    lblFlightTime.Text = $"Flight time: {T} s"
    lblMaxHeight.Text  = $"Max height:  {H} m"
    lblRange.Text      = $"Range:       {R} m"
    lblAngle45.Text    = "Maximum range is achieved at 45°"
End Sub

Copilot Chat Prompts for This Lesson

Try these in the Copilot Chat panel while working with trig functions:

"Write a Sub DrawPolygon(g, cx, cy, sides, radius) that draws a regular polygon using Math.Cos and Math.Sin with 2*PI/sides spacing"
"Implement a radar sweep animation: draw a rotating line using Math.Cos/Sin in a Timer Tick event, and plot blip points"
"Create a Function SoundPressureLevel(distance) that uses Math.Log10 and the inverse square law to compute dB at distance"
"Use the law of cosines (cos C = (a²+b²-c²)/(2ab)) with Math.Acos to solve an oblique triangle given all three sides"

Lesson Summary

All VB 2026 trig functions use radians. Convert with degrees * Math.PI / 180 before passing to Math.Sin, Math.Cos, or Math.Tan. Passing degrees directly produces wrong values silently.
Math.Sin and Math.Cos always return values in [−1, 1]. Math.Tan is undefined at 90° and 270° (±∞). The Pythagorean identity sin²θ + cos²θ = 1 always holds.
Inverse functions: Math.Asin returns [−π/2, π/2]; Math.Acos returns [0, π]; Math.Atan returns (−π/2, π/2). All inputs outside valid domains return Double.NaN.
Use Math.Atan2(y, x) instead of Math.Atan(y/x). Atan2 handles all four quadrants and avoids division-by-zero when x = 0.
Circular motion: x = cx + r * Math.Cos(angle), y = cy + r * Math.Sin(angle). This pattern drives clock hands, orbiting objects, radar sweeps, and regular polygon drawing.
Wave generation: y = A * Math.Sin(2π * f * t + φ). Adjusting amplitude (A), frequency (f), and phase (φ) models sound, oscillations, tides, and AC voltage.

Exercises

Exercise 19.1 — Regular Polygon Drawer

Draw regular polygons (3 to 12 sides) on a PictureBox using a For loop and Math.Cos/Math.Sin
Formula: vertex i at angle 2π*i/n from centre, scaled by radius
Add a NumericUpDown for sides and a TrackBar for rotation angle
Color the interior using a SolidBrush and g.FillPolygon
Copilot challenge: Ask Copilot to "animate the polygon rotating 1 degree per Timer Tick using a running angle variable"

Exercise 19.2 — Projectile Simulator

Build a form with TrackBars for launch speed (10–100 m/s) and angle (1–89°)
Calculate range, max height, and flight time using Math.Sin, Math.Cos, Math.Pow
Draw the parabolic trajectory on a PictureBox sampling 100 time steps
Show that 45° maximises range and add a label when the optimal angle is selected
Copilot challenge: Ask Copilot to "add air resistance: ax = −k*vx, ay = −g − k*vy, using Euler integration in a loop"

Exercise 19.3 — GPS Distance Calculator

Create a form with four TextBoxes for lat/lon of two locations
Implement the Haversine formula using Math.Sin, Math.Cos, Math.Atan2, Math.Sqrt
Also compute the compass bearing (0–360°) from point 1 to point 2
Pre-fill with Malaysian cities: KL, Penang, JB, Kota Kinabalu
Copilot challenge: Ask Copilot to "add a dropdown of preset city pairs and auto-fill the TextBoxes on selection"

Next: Lesson 20 — Format Function

Master VB 2026's string formatting — number formats (C, F, N, P, E), date/time patterns, custom format strings, String.Format, and interpolated strings with format specifiers.

Continue »

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