How to solve sin beta

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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Question

$$\sin\ \beta$$

Differentiate w.r.t. β

$\cos(\beta )$

Steps Using Definition of a Derivative

For a function $f\left(x\right)$, the derivative is the limit of $\frac{f\left(x+h\right)-f\left(x\right)}{h}$ as $h$ goes to $0$, if that limit exists.

$$\frac{\mathrm{d}}{\mathrm{d}\beta }(\sin(\beta ))=\left(\lim_{h\to 0}\frac{\sin(\beta +h)-\sin(\beta )}{h}\right)$$

Use the Sum Formula for Sine.

$$\lim_{h\to 0}\frac{\sin(h+\beta )-\sin(\beta )}{h}$$

Factor out $\sin(\beta )$.

$$\lim_{h\to 0}\frac{\sin(\beta )\left(\cos(h)-1\right)+\cos(\beta )\sin(h)}{h}$$

Rewrite the limit.

$$\left(\lim_{h\to 0}\sin(\beta )\right)\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\left(\lim_{h\to 0}\cos(\beta )\right)\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$

Use the fact that $\beta $ is a constant when computing limits as $h$ goes to $0$.

$$\sin(\beta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\beta )\left(\lim_{h\to 0}\frac{\sin(h)}{h}\right)$$

The limit $\lim_{\beta \to 0}\frac{\sin(\beta )}{\beta }$ is $1$.

$$\sin(\beta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\beta )$$

To evaluate the limit $\lim_{h\to 0}\frac{\cos(h)-1}{h}$, first multiply the numerator and denominator by $\cos(h)+1$.

$$\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)=\left(\lim_{h\to 0}\frac{\left(\cos(h)-1\right)\left(\cos(h)+1\right)}{h\left(\cos(h)+1\right)}\right)$$

Multiply $\cos(h)+1$ times $\cos(h)-1$.

$$\lim_{h\to 0}\frac{\left(\cos(h)\right)^{2}-1}{h\left(\cos(h)+1\right)}$$

Use the Pythagorean Identity.

$$\lim_{h\to 0}-\frac{\left(\sin(h)\right)^{2}}{h\left(\cos(h)+1\right)}$$

Rewrite the limit.

$$\left(\lim_{h\to 0}-\frac{\sin(h)}{h}\right)\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$

The limit $\lim_{\beta \to 0}\frac{\sin(\beta )}{\beta }$ is $1$.

$$-\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)$$

Use the fact that $\frac{\sin(h)}{\cos(h)+1}$ is continuous at $0$.

$$\left(\lim_{h\to 0}\frac{\sin(h)}{\cos(h)+1}\right)=0$$

Substitute the value $0$ into the expression $\sin(\beta )\left(\lim_{h\to 0}\frac{\cos(h)-1}{h}\right)+\cos(\beta )$.

$$\cos(\beta )$$

Show Solution

Evaluate

$\sin(\beta )$