Why are the functions y= cos x and y = sin(x+90) considered the same function?

[Buddah]

1. Explain why the functions y= cos x and y = sin(x+90) are the same function. Explanation must be detailed include graphs if you wish.

2. Outline why the identities are referred to as Pythagorean identities:
sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc ²θ




HELP PLEASEEEE! thanks in advance

 

[hunt_mat]

What have you done so far?

 

[Buddah]

I figured out the first one, i now its the same because the cos functions is just a shift by 90degrees but i don't know how to explain it.

and am lost for number 2 :(

 

[Square]

2. cosine and sine are the x and y-coordinates in the unit circle. At an angle v you can form a triangle with hypothenuse 1 and catheti x and y.

 

[Barbados_Slim]

Rewrite the second function y=sin(x+90) using the double angle formula
$$sin (\alpha + \beta) = sin {\alpha}cos {\beta} + cos {\alpha}sin {\beta}$$
Then the answer to part one of your question should become apparent.

 

[jhae2.718]

For part 2, what is the Pythagorean Theorem? Can you write it in terms of sine and cosine?

 

[hunt_mat]

For part 2, think of a right angled triangle with hypotenuse length R say and then write down pythagoras' thoeren and the definition of sin and cos and the answer should become apparent.

 

[HallsofIvy]

For the first I would also recommend you draw a right triangle. sin is "opposite side over hypotenuse" and cosine is "near side over hypotenuse". Which side is "opposite" or "near" depends on which angle of the triangle you are using. And what is the relationship between angles in a right triangle?

 

[BloodyFrozen]

cos(x) is the x- coordinate
sin(x) is the y- coordinate

Now relate this to the Pythagorean Theorem

 

[Char. Limit]

More to the point, for a circle of radius r, the cosine of a point on that circle is the x-coordinate over r, or x/r. The sine of a point on that circle is the y-coordinate over r, or y/r.

Now, knowing this, and the fact that sin²(z) + cos²(z) = 1, can you show the Pythagorean theorem?