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

u ok Step-by-step explanation:...

Why are the values for sine and cosine equal - id451156857, lukeramjitsingh

art3misdiangelo

14.03.2022

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

Step-by-step explanation:

Method 1

From the isosceles right triangle; See diagram in attachment.

$\sin(45\degree)=\frac{x}{x\sqrt{2} }$

$\sin(45\degree)=\frac{1}{\sqrt{2} }$

$\sin(45\degree)=\frac{\sqrt{2}}{2 }$

Also;

$\sin(45\degree)=\frac{x}{x\sqrt{2} }$

$\cos(45\degree)=\frac{1}{\sqrt{2} }$

$\cos(45\degree)=\frac{\sqrt{2}}{2 }$

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

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

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

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