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Sine & cosine of complementary angles

Learn about the relationship between the sine & cosine of complementary angles, which are angles who together sum up to 90°.
We want to prove that the sine of an angle equals the cosine of its complement.
sine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis
Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90degrees.
Now here's the cool part. See how the sine of one acute angle
describes the start color #11accd, start text, e, x, a, c, t, space, s, a, m, e, space, r, a, t, i, o, end text, end color #11accd as the cosine of the other acute angle?
Incredible! Both functions, sine, left parenthesis, theta, right parenthesis and cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis, give the exact same side ratio in a right triangle.
And we're done! We've shown that sine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis.
In other words, the sine of an angle equals the cosine of its complement.
Well, technically we've only shown this for angles between 0degrees and 90degrees. To make our proof work for all angles, we'd need to move beyond right triangle trigonometry into the world of unit circle trigonometry, but that's a task for another time.

Cofunctions

You may have noticed that the words sine and cosine sound similar. That's because they're cofunctions! The way cofunctions work is exactly what you saw above. In general, if f and g are cofunctions, then
f, left parenthesis, theta, right parenthesis, equals, g, left parenthesis, 90, degrees, minus, theta, right parenthesis
and
g, left parenthesis, theta, right parenthesis, equals, f, left parenthesis, 90, degrees, minus, theta, right parenthesis.
Here is a full list of the basic trigonometric cofunctions:
Cofunctions
Sine and cosinesine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis
cosine, left parenthesis, theta, right parenthesis, equals, sine, left parenthesis, 90, degrees, minus, theta, right parenthesis
Tangent and cotangenttangent, left parenthesis, theta, right parenthesis, equals, cotangent, left parenthesis, 90, degrees, minus, theta, right parenthesis
cotangent, left parenthesis, theta, right parenthesis, equals, tangent, left parenthesis, 90, degrees, minus, theta, right parenthesis
Secant and cosecant\sec, left parenthesis, theta, right parenthesis, equals, \csc, left parenthesis, 90, degrees, minus, theta, right parenthesis
\csc, left parenthesis, theta, right parenthesis, equals, \sec, left parenthesis, 90, degrees, minus, theta, right parenthesis
Neat! Whoever named the trig functions must have deeply understood the relationships between them.

Want to join the conversation?

  • blobby green style avatar for user Mike Mahaney
    how can you study more effectively to do better on tests?
    (3 votes)
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    • leaf orange style avatar for user Tanner Garner
      Understanding the intuition is the key, as Sal likes to remind us of. You will almost always forget formulas, so learn how to derive the formulas; where they come from.
      If you have a hard time remembering the two special triangles, remember, it can all be derived using the Pythagorean Theorem.
      Also, see if you can find out what the material will be on the test and study that. I'm sure your teacher will be thrilled if you ask.
      (8 votes)
  • duskpin seed style avatar for user SC
    So, what kind of questions are related to these confunctions? Im kind of confused because i don't know when to use this
    (6 votes)
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    • blobby blue style avatar for user Jayatra Saxena
      look sarah when you have to find an hypotenuse knowing an angle(x) and the opposite side you will form an equation like

      sin(x)=opp/hyp
      therefore sin(x)/opp=1/hyp
      therefore opp/sin(x)=hyp

      but with cosec funtion you will do it like
      cosec(x)=hyp/opp
      therefore opp*cosec(x)=hyp

      I just wanna conclude saying that cofunctions are helpful when you are giving a math exam without a calculator. Oherwise it is just a concept you should keep in back of your mind.

      Keep it simple just remember that

      cot or cotan=1/tan(x)

      sec or secant = 1/cos(x)

      cosec or cosecant=1/sin(x)

      I can also give you trick to memorize the trig table for angles 0-90 degrees. Please tell me wether my exaination was clear
      (2 votes)
  • blobby green style avatar for user asmaa adam
    how do you find other angles with only the sides given?
    (4 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      With only the sides given, you'd have to solve for an angle using the law of cosines. If the triangle had a right angle, you could use the inverse trig functions. The law of cosines is:
      c^2 = a^2 + b^2 - 2*a*c*cos(C)
      a, b, and c are sides of a triangle, and C is the angle included between a and b. The law of cosines works by imagining and altitude in the triangle, and basing calculations off of it in such a way that you don't need the altitude measurement to solve the triangle, you just need either all three sides, as in your question, or two sides and the included angle.
      (2 votes)
  • blobby green style avatar for user philipoheneatao
    What secant cotangent and cosecant about
    (3 votes)
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  • blobby green style avatar for user thejasonwu8
    What is Secant?
    (2 votes)
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  • blobby green style avatar for user aq9818
    So does sin(θ)=cos(90∘ + theta)?
    (3 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      sin(𝜃) = cos(90° + 𝜃) is typically not true.

      Using the angle addition formula for cosine, we get
      sin(𝜃) = cos(90°) cos(𝜃) − sin(90°) sin 𝜃,

      which simplifies to
      sin(𝜃) = −sin(𝜃) ⇒ sin(𝜃) = 0 ⇒ 𝜃 = 𝑛𝜋, where 𝑛 is an integer.
      (2 votes)
  • boggle blue style avatar for user Bryan
    Do these cofunctions always hold even when theta isn't acute? e.g., sin(1000 degrees) = cos(90 - 1000 degrees)
    (2 votes)
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  • duskpin ultimate style avatar for user 𝔸𝕤𝕥𝕒𝕣𝔾𝕒𝕝𝕒𝕩𝕚𝕖𝕤
    i still am confused on the topic...
    (2 votes)
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  • leaf green style avatar for user Emraan
    Why it can not be Sine@=Csc (90-@)
    And
    Why it is it Sine@=Cos(90-@)*bold
    (2 votes)
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    • duskpin tree style avatar for user VikingTheDude
      sin stands for sine. cos stands for cosine. cosine is the co-function of sine, which is why it is called that way (there's a 'co' written in front of 'sine'). Co-functions have the relationship
      sin@ = cos(90-@)
      However, the trig function csc stands for cosecant which is completely different from cosine. As you might have noticed, cosecant has a 'co' written in front of ''secant'. So we can see here that cosecant is the co-function of secant. Similarly, cotangent is the co-function of tangent. Remember that 'co' in front of those names and it'll be much easier to remember them, so that you do not mistaken csc for cos.
      A good way to remember is to say the entire name to yourself whenever you're doing trigonometry. For example, say 'cosine' instead of just 'cos' and 'tangent' instead of just 'tan'.
      (2 votes)
  • winston default style avatar for user Parthiv Chandra Gajula
    How do I insert "COT" onto my calculator? is it the same as inverse of tangent? where i put "1" as the numerator and whichever trigonometric I am working with?
    (1 vote)
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    • hopper cool style avatar for user obiwan kenobi
      cot(x) = 1/(tan x). If you want to find the cotangent of an angle. Just do 1 divided by the tangent of that angle. Notice, that the words inverse and reciprocal cannot be used interchangeably in trigonometry. cotangent is the RECIPROCAL function of tangent, not its inverse. Hope this helps!
      (3 votes)