Convert Cos Equation to Sine Calculator

Calculator Input

Use the model y = A cos(Bx + C) + D. The result uses sine form with the same graph.

Amplitude A

Example: 3 or -2.5

Frequency Multiplier B

This multiplies the variable.

Phase Constant C

Use the selected angle unit.

Vertical Shift D

Moves the graph up or down.

Variable Symbol

Use x, t, n, or theta.

Angle Unit

Keep one unit throughout.

Amplitude Style

Positive style adds another half turn if needed.

Normalize Final Phase

Keeps the phase inside a cleaner range.

Decimal Precision

Use 0 to 12 decimal places.

Table Start x

Table Step

Table Rows

Formula Used

y = A cos(Bx + C) + D

cos(u) = sin(u + 90°)

y = A sin(Bx + C + 90°) + D

For radians, replace 90° with π/2. If a negative amplitude is forced positive, the calculator adds another 180°, or π radians, to keep the equation equal.

How to Use This Calculator

Enter the amplitude value as A.
Enter the variable multiplier as B.
Enter the phase constant as C.
Enter the vertical shift as D.
Select degrees or radians.
Choose the phase and amplitude display options.
Press the convert button.
Review the table, then download CSV or PDF.

Example Data Table

Cosine Equation	Sine Equivalent	Added Phase	Note
y = 4 cos(x + 30°)	y = 4 sin(x + 120°)	90°	Direct identity
y = -2 cos(3t - 45°) + 1	y = -2 sin(3t + 45°) + 1	90°	Sign kept
y = 5 cos(0.5x) - 2	y = 5 sin(0.5x + 90°) - 2	90°	Zero phase start

Cosine to Sine Conversion Guide

Core Idea

A cosine equation can be rewritten as a sine equation by shifting its angle. The core identity is simple. Cosine equals sine after a quarter turn. That turn is ninety degrees. In radians, it is pi over two.

Equation Parts

This calculator uses the common model y equals A cos(Bx plus C) plus D. The amplitude is A. The frequency multiplier is B. The phase part is C. The vertical shift is D. The calculator keeps these parts visible, so each change can be checked.

Direct Conversion

The direct conversion is A sin(Bx plus C plus ninety degrees) plus D. In radians, the added value is pi over two. This does not change the graph. It only changes the function name and phase position. Every output remains mathematically equal to the original expression.

Advanced Options

Advanced options help avoid common mistakes. You can keep a negative amplitude. You can also force a positive amplitude. When that option is used, another half turn is added to the sine phase. The calculator can also normalize the phase. Normalizing places the angle inside a cleaner standard range.

Value Checks

The value table is useful for proof. It evaluates both equations at matching input values. The difference column should be near zero. Tiny differences can appear from rounding only. This table is helpful for homework, graph checks, and lesson examples.

Unit Choice

Use degrees when your equation uses degree angles. Use radians when your equation uses radian measures. Keep the same unit through the full calculation. Mixing units creates wrong phase shifts. The calculator shows both degree and radian phase changes for reference.

Common Uses

This conversion is often used in trigonometry, signals, waves, and graph transformations. Sine and cosine describe the same wave shape. Their main difference is phase. A clean sine form can make comparison easier. It can also match class notes or software input rules.

Graph Support

A careful result also supports graphing. The sine form starts later or earlier on the horizontal axis. That movement is controlled by the phase. The shape, height, and center line stay unchanged.

Best Practice

For best results, enter exact coefficients first. Then choose the display precision. Review the converted equation. Finally, export the CSV or PDF for records.

FAQs

What does this calculator convert?

It converts equations written as A cos(Bx + C) + D into equal sine equations. The graph stays the same. Only the function name and phase value change.

Which identity is used?

The calculator uses cos(u) = sin(u + 90°). In radians, it uses cos(u) = sin(u + π/2). Both identities create an equivalent sine expression.

Does the amplitude change?

The amplitude normally stays the same. If you select positive amplitude style, a negative amplitude is changed to positive. The phase is adjusted so the equation remains equal.

Should I use degrees or radians?

Use the same unit as your original equation. Degree equations should use degrees. Radian equations should use radians. Mixing units gives an incorrect phase shift.

What does phase normalization do?

Phase normalization rewrites the final phase into a cleaner standard range. It does not change the wave. It only makes the displayed angle easier to read.

Why is the difference column almost zero?

The difference column compares the original cosine value with the converted sine value. Very tiny values can appear because of decimal rounding in computer calculations.

Can this handle vertical shifts?

Yes. Enter the vertical shift as D. The calculator carries D into the sine equation without changing it, because phase conversion affects only the angle.

Can I export the result?

Yes. After converting, use the CSV button for spreadsheet data. Use the PDF button for a printable summary with the formula and value table.