Why does sin(x) = cos(90−x) and sin(90−x) = cos(x)?

[Kristina123]

Why does sin(x) = cos(90−x) and sin(90−x) = cos(x)?

I think it's because both trigonometric ratios use the hypotenuse plus either the adjacent or opposite side. The subtraction part plays a role by subtracting the bigger adjacent/opposite side.

Of course, I know that sin(x) = cos(90−x) because I punched it in the calculator, but I'm getting frustrated and am struggling to understand why. Can someone please better explain it to me with loads of details but simple?

 

[tkhunny]

Find the circle with all the angles marked in it.

Unit circle - Wikipedia

en.wikipedia.org

Look long and hard at the 30º and 60º angles. keeping in mind that 30 = 90 - 60 and 60 = 90 - 30.

 

[Steven G]

It is great that this bothers you! I always have told my students never to believe anything I say to them but rather go home and convince themselves that I am correct.

No need for loads of details. Draw a right triangle ABC with \(\displaystyle \angle B\) being the 90^o angle. Now the other two angles must add up to 90^o (so the sum of all 3 angles is 180^o)
Call one angle x^oand the other angle (90-x)^o. Is this clear?
Now label the 3 sides with a, b and c. What is the sin(x)? What is the cos(90-x)? Also what is the cos(x) and sin(90-x)?

Alternatively expand sin(90-x) and cos(90-x).

Did you get the results you wanted?

 

[Kristina123]

It is great that this bothers you! I always have told my students never to believe anything I say to them but rather go home and convince themselves that I am correct.

No need for loads of details. Draw a right triangle ABC with \(\displaystyle \angle B\) being the 90^o angle. Now the other two angles must add up to 90^o (so the sum of all 3 angles is 180^o)
Call one angle x^oand the other angle (90-x)^o. Is this clear?
Now label the 3 sides with a, b and c. What is the sin(x)? What is the cos(90-x)? Also what is the cos(x) and sin(90-x)?

Alternatively expand sin(90-x) and cos(90-x).

Did you get the results you wanted?

I got the visual representation. But I still don't know why sin(x) = cos(90−x). Can you explain in words, rather than in a visual?

 

[Steven G]

I got the visual representation. But I still don't know why sin(x) = cos(90−x). Can you explain in words, rather than in a visual?

In a right triangle there is only one hypotenuse. The oppose of angle x is the adjacent of angle (90-x). So sin(x) = cos(90-x).
By the way, I am starting to think that this is a homework assignment!

 

[Dr.Peterson]

There are several ways to answer any "why" question; I can't tell which one your textbook would intend, or which would satisfy you (according to which is the source of the question).

One is that this is just the definition! The name cosine, in fact, is short for "the complement's sine". The definition in the right triangle is made to make that true. (You do know that the two acute angles in a right triangle are complements, right?)

 

[JeffM]

I think jomo's explanation is the most intuitive, but perhaps different words and a sketch will help.

Sketch a right triangle. Label the two angles that are not right angles [MATH]\theta[/MATH] and [MATH]\phi[/MATH].

Label the hypotenuse as H, the side opposite [MATH]\theta[/MATH] as O, and the side adjacent to [MATH]\theta[/MATH] as A.

Do you have your sketch?

Is it not obvious looking at the sketch that

[MATH]sin(\theta) = \dfrac{O}{H} = cos(\phi)[/MATH]
and

[MATH]cos(\theta) = \dfrac{A}{H} = sin(\phi)[/MATH]
because the side opposite [MATH]\phi[/MATH] is A and the side adjacent to [MATH]\phi[/MATH] is O.

And [MATH]\theta + \phi + 90 = 180 \implies \theta + \phi = 90 \implies \phi = 90 - \theta.[/MATH]
Consequently,

[MATH]sin(\theta) = cos(\phi) = cos(90 - \theta) \text { and } cos(\theta) = sin(\phi) = sin(90 - \theta).[/MATH]
It really is that simple: what is opposite for one angle is adjacent to the other, and what is adjacent for one angle is opposite to the other.

 

[Kristina123]

In a right triangle there is only one hypotenuse. The oppose of angle x is the adjacent of angle (90-x). So sin(x) = cos(90-x).
By the way, I am starting to think that this is a homework assignment!

Lol. I can definitely tell you this that this is not a homework assignment, nor homework question.

 

[anony-mouse1]

It is great that this bothers you! I always have told my students never to believe anything I say to them but rather go home and convince themselves that I am correct.

No need for loads of details. Draw a right triangle ABC with \(\displaystyle \angle B\) being the 90^o angle. Now the other two angles must add up to 90^o (so the sum of all 3 angles is 180^o)
Call one angle x^oand the other angle (90-x)^o. Is this clear?
Now label the 3 sides with a, b and c. What is the sin(x)? What is the cos(90-x)? Also what is the cos(x) and sin(90-x)?

Alternatively expand sin(90-x) and cos(90-x).

Did you get the results you wanted?

Isn't sin(x) = BC/AC? How does that correlate to angle measures?

 

[Deleted member 4993]

I got the visual representation. But I still don't know why sin(x) = cos(90−x). Can you explain in words, rather than in a visual?

Are you comfortable with the following equations:

cos(α - Θ) = cos(α) * cos(Θ) + sin(Θ) * sin(α)

cos(π/2) = 0 and

sin(π/2) = 1

 

[Dr.Peterson]

Isn't sin(x) = BC/AC? How does that correlate to angle measures?

I'm not sure what you are asking about this year-old question.

If you call angle ACB y, then y = 90-x degrees, and BC/AC = cos(y). So cos(90-x) = sin(x).

 

[Karak10]

I just understood it a while ago.
Like Jomo said, create a triangle ABC, the angle of a should be 90, the other two are uknown, but we know that b + c = 90 since the sum of all the angles should equal 180. Now, assuming that B is at the top, and C is at the right side, then sin(c) should equal AB / BC, you will also notice however that cos(b) equals AB / BC, which means that sin(c) = cos(b), what is b and what is c though? If c = x then b = 90 - x, which means that sin(x) = cos(90 - x). Try drawing the triangle and confirm that what I am saying is true, and this will probably start making sense for you too.