How to convert a Sine Function Into a Cosine

To convert a sine function into a cosine function, shift the sine function by 90° (or π/2 radians) to the left. Mathematically:

sin⁡(x) = cos⁡(x − π/2)

Let's discuss this in detail in this article.

Sine and Cosine Functions

Before diving into the conversion process It is essential to understanding the basic definitions and properties of sine and cosine functions.

Sine of an angle θ in a right angled triangle is the ratio of the length of the opposite side to the hypotenuse i.e.,

sin θ = Opposite Side/Hypotenuse

The cosine of an angle \theta is the ratio of the length of the adjacent side to the hypotenuse.

cos θ = Adjacent Side/Hypotenuse

Relationship Between Sine and Cosine Functions

Sine and cosine functions are closely related and differ only by a phase shift. Cosine leads sine by 90 degrees (or π/2 radians), meaning:

sin⁡(x) = cos⁡(x − π/2) and cos(x) = sin⁡(π/2 + x)

They also share the same amplitude and period.

We can also use Pythagorean identity i.e., sin⁡²(x) + cos⁡²(x) = 1 to express sine in terms of cosine and cosine in terms of sine as follows:

\sin(x) = \sqrt{1 - \cos^2(x)}

\cos(x) = \sqrt{1 - \sin^2(x)}

In both cases, use the positive or negative root depending on the quadrant in which the angle lies.

Steps to Convert a Sine Function into a Cosine Function

To convert a sine function into a cosine function follow these steps

• Step 1: Identify the Sine Function

Start with the sine function you want to convert such as sin θ.

• Step 2: Apply the Phase Shift

Use the relationship sin⁡ θ = cos⁡(θ − π/2)

• Step 3: Simplify the Expression

Let's consider some examples for better understanding.

Example 1: Convert sin (π/4) to a cosine function.

Solution:

Given: sin (π/4)

Using identity sin⁡ θ = cos⁡(θ − π/2)

sin⁡ π/4 = cos⁡(π/4 − π/2) = cos (-π/4) = cos (π/4)

[As cos(-x) = cos x]

Example 2: Express sin (2x - π) as a cosine function.

Solution:

Given: sin (2x - π)

Using identity sin⁡ θ = cos⁡(θ − π/2)

sin (2x - π) = cos (2x - π - π/2) = cos (2x - 3π/2)

Read More,

• Sine Function
• Cosine Function
• Trigonometry
• Trigonometric Identities
• Pythagorean Identities