Featured 1 month ago

One does not use integration or rationalizing to prove this. One proves this by changing only 1 side of the equation, until it is identical to the other side.

Prove:

The identity for

Use the fact that

Write

Multiply the fraction inside the square by 1 in the form of

Perform the multiplication:

Please observe the cancellation of the embedded denominators:

Write the equation without the cancelled factors:

Perform the multiplication implied by the square:

Use the identity

Use the identity

Featured 1 month ago

Please see below.

To sketch the angle in standard position,

One side of the angle is the positive

The terminal side has one end at the origin (the point

Now draw a line from the origin through the point

If we knew how the angle was made (which direction and how many times around the circle), we would show that also.

Give the angle a name. I will use

**Memorize this**

If the point

The angle has sine

**For this question**

We have

So the sine of

Many trigonometry teachers will insist that you write your answer with a rational number in the denominator. If this is something you have to do, multiply the answer by

So we should answer

The cosine of

Our answer is

Featured 4 weeks ago

**meaningless,**

Remembering that,

and, adding these give,

Obviously, **no** such

Hence,

**Enjoy Maths.!**

Featured 2 weeks ago

Draw a sketch first and fill in what you know.

Start with a rectangle and draw in one diagonal.

The jetliner is at a vertical height of

This is the breadth of the rectangle.

The jetliner is at the top of the rectangle where the diagonal starts.

The diagonal is the direct line from the jetliner to the coast ahead and below, and is the hypotenuse of a right-angled triangle.

The angle of depression is the angle between the horizontal length and the diagonal. It can be shown in two places, because the angle of elevation is equal to the angle of depressions, because they are alternate angles on parallel lines.

These angles are always measured FROM THE HORIZONTAL!

We need to find the length of the rectangle.

Now to convert to miles:

Featured 2 weeks ago

graph{3sin(2(x-1)) [-10, 10, -5, 5]}

If we consider

graph{3sinx [-10, 10, -5, 5]}

We will look at C next, this is the movement of the graph left or right, where a negative C value moves the graph to the right. So we move the whole graph 1 to the right in this case.

graph{3sin(x-1) [-10, 10, -5, 5]}

Finally B is stretching the graph parallel to the x axis by a factor of

So in your case B = 2, so

Then graphing this:

graph{3sin(2(x-1)) [-10, 10, -5, 5]}

Featured 4 days ago

As

Further, as

Hence

=

and

=

and

=

=

Hence,

Featured 1 week ago

In fact,

We use the following property known as **Napier's Rule (NR) :**

It can be proved very easily using the **Sine Rule** for

In our **Problem,** we have,

By the **Componedo-Dividendo,** we get,

**Foot Note :** In fact,

Solving,

**Enjoy Maths.!**

Featured 4 days ago

This is essentially a guess, but note that all of these angles have exact algebraic expressions for the their trigonometric values. So the question probably wants you to express this quotient as an exact algebraic expression.

Since all of the angles are positive but less than

To find the value of

Then:

#sin 45^@ = "opposite"/"hypotenuse" = 1/sqrt(2) = sqrt(2)/2#

For the other two trigonometric values, consider a right angled triangle with sides

We have:

#cos 30^@ = "adjacent"/"hypotenuse" = sqrt(3)/2#

#sin 60^@ = "opposite"/"hypotenuse" = sqrt(3)/2#

Having obtained our trigonometric values, here's the algebra:

#sin 45^@/(cos 30^@+sin 60^@) = (sqrt(2)/2)/(sqrt(3)/2+sqrt(3)/2)#

#color(white)(sin 45^@/(cos 30^@+sin 60^@)) = sqrt(2)/(2sqrt(3))#

#color(white)(sin 45^@/(cos 30^@+sin 60^@)) = (sqrt(2)sqrt(3))/6#

#color(white)(sin 45^@/(cos 30^@+sin 60^@)) = sqrt(6)/6#

Featured 1 week ago

In order to answer this I have assumed a vertical shift of

The standard cos function

If we want a period of

That is

To get an amplitude of

There is to be no horizontal shift, so the argument for

In order to achieve the vertical shift (that I assumed would be

Featured 4 days ago

When

when

Hence combining we have