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Sine & cosine of complementary angles

Sal shows that the sine of any angle is equal to the cosine of its complementary angle. Created by Sal Khan.

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  • male robot hal style avatar for user Anuraag Warudkar
    I tried the operation on a calculator and found that:
    sin(32) is about 0.55
    cos(58) is about 0.12
    What am i missing?
    (6 votes)
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  • blobby green style avatar for user wr20398
    Just to clarify an SAT question, is sin(xº) always equal to cos(90º-xº)?

    ***SPOILER: below is one of the Khan academy practice test questions***

    In a right triangle, one angle measures xº where sin(xº) = 4/5. What is cos(90º - xº)?
    (8 votes)
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  • winston default style avatar for user dacanine
    What is an arbitrary angle?
    (5 votes)
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  • marcimus purple style avatar for user emily
    What is the difference between radians and degrees?
    (4 votes)
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    • leaf green style avatar for user kubleeka
      They're just different units to measure the same quantity (angle measure), like how pounds and kilograms are different units for the same quantiy (mass). A degree is defined so that there are 360 degrees in a full circle, and a radian is defined so that there are 2π radians in a full circle.
      (9 votes)
  • blobby green style avatar for user Happy.kima
    can someone please help me i feel dumb :(
    (5 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      Well, we know that the sine of an angle is the ratio of the opposite to hypotenuse. Similarly, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. If we pause and imagine a right triangle, the sine of one angle would be the cosine of the angle across from it, since the hypotenuse is constant, but the opposite side of one angle and the adjacent side of the other angle refer to the same side. Since we are talking about a right triangle, the angles are complementary. And this fact gives us enough information to conclude the following equation:
      sin(x degrees) = cos(90 - x degrees), and vice versa.
      If you have any further questions, please leave them in a comment, and I'll get right to them!
      (7 votes)
  • blobby green style avatar for user Roshini
    I have a question on this concept:

    If sin(x) = cos(90 - x)

    then how do I find the x in
    sin(50 - x) = cos(3x + 10)
    (3 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      You could rearrange the concept a bit to get that the sum of the arguments must be 90 degrees for the sides to be equal, since the sine is the same as the cosine of the complementary angle. We can then set up an equation with just the arguments:
      50 - x + 3x + 10 = 90
      2x + 60 = 90
      2x = 30
      x = 15
      (7 votes)
  • sneak peak green style avatar for user fresh_cutz
    What is near that 90° - ø thing?
    It's the (ø) that triggers me.
    (2 votes)
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    • female robot ada style avatar for user Angelica Chen
      It's actually θ, the Greek letter theta. Lowercase greek letters are commonly used to represent angle measures. You might also see alpha, which looks like an a, as well as many others. So that you don't get lost, here is a copy of the Greek alphabet: ςερτυθιοπασδφγηξκλζχψωβνμ.

      If you look closely, you'll see that π is in there, since it is also a Greek letter.

      Don't get too confused, these work the same way as any other variable, like a, b, x, and y.

      Hope that helps!
      (6 votes)
  • aqualine seed style avatar for user Scarlet Castillo
    This is a really cool concept! But how does this help you solve a triangle? I mean what sort of question would this work on?
    (3 votes)
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  • starky ultimate style avatar for user Samith Hasan
    At what does he mean by theta
    (1 vote)
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  • purple pi purple style avatar for user Tobia
    I've got two silly simple questions:
    1) From the definition of sine, cosine and tangent it turns out that the tangent of an angle is equal to sin (angle) / cos (angle), so sin/cos = tangent, and maybe that is just a stupid thing i noticed...
    2) Should i write tan or tg for the tangent?
    (1 vote)
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    • piceratops ultimate style avatar for user Just Keith
      1) It is not a stupid thing, that tan x = sin x / cos x is a VERY important equality. It will be used extensively should you move on to study calculus. So, it is quite impressive that you notice this. Also, unlike the other "definitions" of the trig functions that are usually given at this level of study, tan x = sin x /cos x is always true for all angles, not just right triangles. You will note that tangent is undefined at values of x where cos x = 0.

      2) The standard symbol for tangent is "tan". Do not use "tg" as that has other meanings.
      (3 votes)

Video transcript

We see in a triangle, or I guess we know in a triangle, there's three angles. And if we're talking about a right triangle, like the one that I've drawn here, one of them is going to be a right angle. And so we have two other angles to deal with. And what I want to explore in this video is the relationship between the sine of one of these angles and the cosine of the other, the cosine of one of these angles and the sine of the other. So to do that, let's just say that this angle-- I guess we could call it angle A-- let's say it's equal to theta. If this is equal to theta, if it's measure is equal to theta degrees, say, what is the measure of angle B going to be? Well, the thing that will jump out at you-- and we've looked at this in other problems-- is the sum of the angles of a triangle are going to be 180 degrees. And this one right over here, it's a right triangle. So this right angle takes up 90 of those 180 degrees. So you have 90 degrees left. So these two are going to have to add up to 90 degrees. This one and this one, angle A and angle B, are going to be complements of each other. They're going to be complementary. Or another way of thinking about it is B could be written as 90 minus theta. If you add theta to 90 degrees minus theta, you're going to get 90 degrees. Now, why is this interesting? Well, let's think about what the sine of theta is equal to. Sine is opposite over hypotenuse. The opposite side is BC. So this is going to be the length of BC over the hypotenuse. The hypotenuse is side AB. So the length of BC over the length of AB. Now, what is that ratio if we were to look at this angle right over here? Well, for angle B, BC is the adjacent side, and AB is the hypotenuse. From angle B's perspective, this is the adjacent over the hypotenuse. Now, what trig ratio is adjacent over hypotenuse? Well, that's cosine. Sohcahtoa, let me write that down. Doesn't hurt. Sine is opposite over hypotenuse. We see that right over there. Cosine is adjacent over hypotenuse, cah. And toa, tangent is opposite over adjacent. So from this angle's perspective, taking the length of BC, BC is its adjacent side, and the hypotenuse is still AB. So from this angle's perspective, this is adjacent over hypotenuse. Or another way of thinking about it, it's the cosine of this angle. So that's going to be equal to the cosine of 90 degrees minus theta. That's a pretty neat relationship. The sine of an angle is equal to the cosine of its complement. So one way to think about it, the sine of-- we could just pick any arbitrary angle-- let's say, the sine of 60 degrees is going to be equal to the cosine of what? And I encourage you to pause the video and think about it. Well, it's going to be the cosine of 90 minus 60. It's going to be the cosine of 30 degrees. 30 plus 60 is 90. And of course, you could go the other way around. We could think about the cosine of theta. The cosine of theta is going to be equal to the adjacent side to theta, to angle A, I should say. And so the adjacent side is right over here. That's AC. So it's going to be AC over the hypotenuse, adjacent over hypotenuse. The hypotenuse is AB. But what is this ratio from angle B's point of view? Well, the sine of angle B is going to be its opposite side, AC, over the hypotenuse, AB. So this right over here, from angle B's perspective, this is angle B's sine. So this is equal to the sine of 90 degrees minus theta. So the cosine of an angle is equal to the sine of its complement. The sine of an angle is equal to the cosine of its complement.