Question

Why are the values for sine and cosine equal at 45°?

Answer 1

The complement of
`45\degree` is
`45\degree`.

Explanation:

Method 1

From the isosceles right triangle; See diagram in attachment.

`\sin(45\degree)=(x)/(x√(2) )`

`\sin(45\degree)=(1)/(√(2) )`

`\sin(45\degree)=(√(2))/(2 )`

Also;

`\sin(45\degree)=(x)/(x√(2) )`

`\cos(45\degree)=(1)/(√(2) )`

`\cos(45\degree)=(√(2))/(2 )`

Hence;

`\cos(45\degree)=\sin(45\degree)`

Method 2

`\sin(45\degree)=\cos(45\degree)` because the complement of
`45\degree` is still
`45\degree`.

Complementary angles add to
`90\degree`.

In general if we have an angle
`\theta`, then its complement is
`(90-\theta)\degree`.

There is a relationship between the sine of an angle,
`\theta` and the cosine of its complement,(90-\theta)\degree[/tex].

The relationship is that,

`\sin(\theta \degree)=\cos((90-\theta)\degree)`.

If
`\theta=45\degree`, then

`\sin(45 \degree)=\cos((90-45)\degree)`.

`\Rightarrow \sin(45\degree)=\cos(45\degree)`.

Answer 2

For this case we have a rectangle triangle.
The opposite leg is the sine of the angle.
The adjacent catheto is the cosine of the angle.
For 45 degrees, the length of both legs is the same.
Thus,
Sine (45) = Cosine (45)
Answer:
The length of the legs is the same for a 45 degree angle.