# Basic calculus 1 question: What is the defining formula of sinΘ?

In summary, Derive the derivative of sinΘ using the definition of differentiation. You will need the limit for cos(h) - 1/h and the identity for the sine of the sum of two angles. You will also need a limit: \lim_{h \to 0} \frac{cos(h) - 1}{h} and the value of this limit.

## Homework Statement

Q) Derive the dereitvative of sinΘ using the definition of differentiation.

I am in Calculus 3, and I used to know how to work this problem very well! :) I just don't remember lol - I just need a little help I guess! :)

## Homework Equations

lim [f(x+h) - f(x))]/h
x→0

## The Attempt at a Solution

f(x) = sinΘ

f(x+h) = (sinΘ + h)

Now I am STUCK! lol

well now put that into the formula and check the numerator.

You will have sin(θ+h)-sinθ. Do you know your sum to product formulas? These will help greatly here or you can expand sin(θ+h) as well.

## Homework Statement

Q) Derive the dereitvative of sinΘ using the definition of differentiation.

I am in Calculus 3, and I used to know how to work this problem very well! :) I just don't remember lol - I just need a little help I guess! :)

## Homework Equations

lim [f(x+h) - f(x))]/h
x→0

## The Attempt at a Solution

f(x) = sinΘ

f(x+h) = (sinΘ + h)

Now I am STUCK! lol

Might as well use x instead of theta, since x is easier to type.

f(x + h) = sin(x + h)
Use the identity for the sine of the sum of two angles.

You will also need a limit:
$$\lim_{h \to 0} \frac{sin(h)}{h}$$
Do you remember the value of this limit?

so, I write:

lim [sin(x+h) - sin(x))]/h
x→0

So, I get:

lim [sin(x) cos(h) + cos(x)sin(h) - sin(x))]/h
x→0

But now I am stuck again lol - I understand that if I use $$\lim_{h \to 0} \frac{sin(h)}{h}$$

then my equation should become:

lim [sin(x) cos(h) + cos(x)sin(h) ]/h , right?
x→0

Your limit should be as h --> 0, not as x --> 0. Also, you are missing a term in the numerator. What happened to the -sin(x) that used to be there?

There's another limit that you'll need as well:
$$\lim_{h \to 0} \frac{cos(h) - 1}{h}$$

Took me a while for me to figure this out! I spent time expanding on the ideas both of you gave me and finally understood what to do!

THANKS to BOTH of you! :)

## 1. What is the general formula for calculating a sine value?

The general formula for calculating a sine value is sinΘ = opposite/hypotenuse, where Θ represents the angle in a right triangle.

## 2. What is the unit circle and how is it related to the sine function?

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It is related to the sine function because the value of sine at any angle on the unit circle is equal to the y-coordinate of that point.

## 3. How is the sine function used in calculus?

The sine function is used in calculus to represent the rate of change of a periodic function. It is also used to calculate the area under a curve in trigonometric integrals.

## 4. Can the sine function be negative?

Yes, the sine function can be negative. The sine of an angle can be positive, negative, or zero, depending on the quadrant in which the angle lies.

## 5. How can the sine function be used to solve real-world problems?

The sine function can be used to solve real-world problems involving periodic phenomena, such as sound waves, ocean tides, and alternating current circuits. It can also be used in navigation and engineering to calculate distances and angles.

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