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### Course: Geometry (FL B.E.S.T.)>Unit 7

Lesson 8: Sine & cosine of complementary angles

# 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.

## Want to join the conversation?

• I tried the operation on a calculator and found that:
What am i missing?
• Your calculator needs to be set in degree mode. Then they will be equivalent measurements.
• 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)?
• Well I'm answering it 4 years after it has been asked but anyways,...
we know that, sin(x)=cos(90-x)
and we have sin(x)=4/5
so we can deduce from above equations that
cos(90-x)=4/5.. hope it helps..
• What is an arbitrary angle?
• That means what he's saying applies to any angle regardless of measure.
• 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!
• What is near that 90° - ø thing?
It's the (ø) that triggers me.
• 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!
• What is the difference between radians and degrees?
• 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.
• 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)
• 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
• im gonna be honest i dont understand a single thing i don't even understand how he did it
• If you think about it in terms of a right triangle, you can have angles and opposite sides, let C be the right angle and c be the hypotenuse. Then you have angle A and side opposite a and angle B and side opposite b. The sin(A)=opp/hyp ]=a/c and the cos(A)=adj/hyp=b/c. Also, the sin(B)=b/c and cos(B)=a/c. We also know that with a right triangle, the two acute angles have to add up to 90 degrees (or complementary) M<A+M<B=90 (thus m<A=90-m<B and m<B-=90-m<A).. Looking at this, sin(A)=cos(B) and cos(A)=sin(B), but with substitution, you could also say sin(A)=cos(90-A) and cos(A)=sin(90-A).
• 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?