Download Sine, cosine, and cofunctions

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Subject: Trigonometry 
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 Sine, cosine, and cofunctions
We want to prove that the sine of an angle equals the cosine of its complement:
sin(θ) = cos(90∘ − θ)
[I'm skeptical. Please show me an example.]
Let's start with a right triangle, and notice how the acute angles are complementary (sum to 90∘ ).
θ
90 − θ
[Help! Please break this down for me.]
Now here's the cool part. See how the sine of one acute angle...
θ
h
90 − θ
sin(θ) =
l
h
l
...describes the exact same ratio as the cosine of the other acute angle:
θ
h
90 − θ
cos(90 − θ) =
l
h
l
Incredible! Both functions, sin(θ) and cos(90∘
− θ), give the exact same side ratio in a right triangle.
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Saweet!
And we're done! We've shown that
sin(θ) = cos(90∘ − θ).
In other words, the sine of an angle equals the cosine of its compliment.
Well, technically we've only shown this for angles between 0∘ and 90∘ . 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 (θ) = g(90∘ − θ)
and
g(θ) = f (90∘ − θ).
Here is a full list of the basic trigonometric cofunctions:
Cofunctions
Sine and cosine
sin(θ) = cos(90∘ − θ)
cos(θ) = sin(90∘ − θ)
Tangent and cotangent
tan(θ) = cot(90∘ − θ)
cot(θ) = tan(90∘ − θ)
Secant and cosecant
sec(θ) = csc(90∘ − θ)
csc(θ) = sec(90∘ − θ)
Neat! Whoever named the trig functions must have deeply understood the relationships between them.
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