sin(pi-x) Formula | Formula of sin(pi-theta)

 For any x, sin(pi-x) formula is given below:

sin(pi-x)=sinx

Similarly, sin(pi-theta)=sin theta. In this post, we will learn how to compute sin(pi-x) or sin(pi-theta).

sin(pi-x) Simplify

To find the formula of sin(π-x), we will apply the following trigonometric identity:

sin(A-B) = sinA cosB – cosA sinB …(∗)

Here we put A=π and B=x; so that we get

sin(π-x) = sinπ cosx – cosπ sinx ⇒ sin(π-x) = 0⋅cosx – (-1) sinx as we know that sinπ=0 and cosπ=-1. ⇒ sin(π-x) = sinx.

Thus we have established the formula of sin(π-x) which is given by

sin(π-x) = sinx

Note: In a similar way, we obtain the formula of sin(π-θ) which is given by sin(π-θ) = sinθ.

More Readings: How to Simplify cos(π-x)

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General Solution of sinx=1 | General Solution of sinx=-1

Simplify sin(x-pi)

The formula of sin(x-π) is given by

sin(x-π) = -sinx.

To prove it, we will put A=x, B=π in (*). Thus, we obtain that

sin(x-π) = sinx cosπ – cosx sinπ ⇒ sin(x-π = sinx⋅ (-1) – cosx ⋅ 0 ⇒ sin(x-π = -sinx.

Hence, we obtain the formula of sin(x-π) which is equal to sin(x-π) = -sinx.

Question Answer

Question: Find the value of sin(-150°)

Answer:

We have sin(-150°) = sin(30°-180°)

So sin(-150°) = sin(30°-π)

⇒ sin(-150°) = -sin30° using the above formula sin(x-π) = -sinx with x=30°.

⇒ sin(-150°) = -1/2 as the value of sin30 is 1/2.

So the value of sin negative 150 is equal to 1/2

Also Read:

How to simplify cos(sin^-1x)

1-sec²x Formula | 1+cot²x Formula

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