cot ((x)/(2))=a=>sin x= ? (cot: ctg: cotangent)

Express cot(x/2): Express cot(x/2) in terms of sine and cosine. Cotangent is the reciprocal of tangent, which is sine over cosine. Therefore, cot(x/2) can be written as cos(x/2)/sin(x/2).Use Pythagorean identity: Use the Pythagorean identity to express sin(x/2) in terms of cos(x/2). The Pythagorean identity states that sin^2(θ) + cos^2(θ) = 1. We can solve for sin(x/2) by rearranging the identity to sin^2(x/2) = 1 - cos^2(x/2) and then taking the square root.Express sin(x): Express sin(x) in terms of sin(x/2) and cos(x/2) using the double angle formula. The double angle formula for sine is sin(x) = 2 * sin(x/2) * cos(x/2).Substitute into formula: Substitute the expression for cot(x/2) into the double angle formula. Since cot(x/2) = cos(x/2)/sin(x/2), we can write sin(x) as sin(x) = 2 * sin(x/2) * (1/sin(x/2)) * cot(x/2).Simplify sin(x): Simplify the expression for sin(x). The sin(x/2) terms cancel out, leaving us with sin(x) = 2 * cot(x/2). Since cot(x/2) is given as a, we have sin(x) = 2a.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

cot ((x)/(2))=a=>sin x= ?
(cot: ctg: cotangent)

$\cot \frac{x}{2}=a \Rightarrow \sin x=$ ? $\newline$ (cot: ctg: cotangent)

View step-by-step help

Home

Math Problems

Algebra 2

Sin, cos, and tan of special angles

Full solution

\cot \frac{x}{2}=a \Rightarrow \sin x=

\newline

(cot: ctg: cotangent)

Express $\cot\left(\frac{x}{2}\right)$ : Express $\cot\left(\frac{x}{2}\right)$ in terms of sine and cosine. $\newline$ Cotangent is the reciprocal of tangent, which is sine over cosine. Therefore, $\cot\left(\frac{x}{2}\right)$ can be written as $\frac{\cos\left(\frac{x}{2}\right)}{\sin\left(\frac{x}{2}\right)}$ .
Use Pythagorean identity: Use the Pythagorean identity to express $\sin(\frac{x}{2})$ in terms of $\cos(\frac{x}{2})$ . The Pythagorean identity states that $\sin^2(\theta) + \cos^2(\theta) = 1$ . We can solve for $\sin(\frac{x}{2})$ by rearranging the identity to $\sin^2(\frac{x}{2}) = 1 - \cos^2(\frac{x}{2})$ and then taking the square root.
Express $\sin(x)$ : Express $\sin(x)$ in terms of $\sin(\frac{x}{2})$ and $\cos(\frac{x}{2})$ using the double angle formula. $\newline$ The double angle formula for sine is $\sin(x) = 2 \cdot \sin(\frac{x}{2}) \cdot \cos(\frac{x}{2})$ .
Substitute into formula: Substitute the expression for $\cot(\frac{x}{2})$ into the double angle formula. $\newline$ Since $\cot(\frac{x}{2}) = \frac{\cos(\frac{x}{2})}{\sin(\frac{x}{2})}$ , we can write $\sin(x)$ as $\sin(x) = 2 \cdot \sin(\frac{x}{2}) \cdot \left(\frac{1}{\sin(\frac{x}{2})}\right) \cdot \cot(\frac{x}{2})$ .
Simplify $\sin(x)$ : Simplify the expression for $\sin(x)$ . The $\sin\left(\frac{x}{2}\right)$ terms cancel out, leaving us with $\sin(x) = 2 \cdot \cot\left(\frac{x}{2}\right)$ . Since $\cot\left(\frac{x}{2}\right)$ is given as $a$ , we have $\sin(x) = 2a$ .