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### Course: Trigonometry>Unit 1

Lesson 7: The reciprocal trigonometric ratios

# 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.
$\mathrm{sin}\left(\theta \right)=\mathrm{cos}\left({90}^{\circ }-\theta \right)$
Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90${}^{\circ }$.
Now here's the cool part. See how the sine of one acute angle
describes the $\text{exact same ratio}$ as the cosine of the other acute angle?
Incredible! Both functions, $\mathrm{sin}\left(\theta \right)$ and $\mathrm{cos}\left({90}^{\circ }-\theta \right)$, give the exact same side ratio in a right triangle.
And we're done! We've shown that $\mathrm{sin}\left(\theta \right)=\mathrm{cos}\left({90}^{\circ }-\theta \right)$.
In other words, the sine of an angle equals the cosine of its complement.
Well, technically we've only shown this for angles between 0${}^{\circ }$ and 90${}^{\circ }$. 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
$f\left(\theta \right)=g\left({90}^{\circ }-\theta \right)$
and
$g\left(\theta \right)=f\left({90}^{\circ }-\theta \right)$.
Here is a full list of the basic trigonometric cofunctions:
Cofunctions
Sine and cosine$\mathrm{sin}\left(\theta \right)=\mathrm{cos}\left({90}^{\circ }-\theta \right)$
$\mathrm{cos}\left(\theta \right)=\mathrm{sin}\left({90}^{\circ }-\theta \right)$
Tangent and cotangent$\mathrm{tan}\left(\theta \right)=\mathrm{cot}\left({90}^{\circ }-\theta \right)$
$\mathrm{cot}\left(\theta \right)=\mathrm{tan}\left({90}^{\circ }-\theta \right)$
Secant and cosecant$\mathrm{sec}\left(\theta \right)=\mathrm{csc}\left({90}^{\circ }-\theta \right)$
$\mathrm{csc}\left(\theta \right)=\mathrm{sec}\left({90}^{\circ }-\theta \right)$
Neat! Whoever named the trig functions must have deeply understood the relationships between them.

## Want to join the conversation?

• how can you study more effectively to do better on tests?
• I'd say just keep practicing problems and remember Soh Cah Toa. I'd also memorize the two triangles.
• So, what kind of questions are related to these confunctions? Im kind of confused because i don't know when to use this
• 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
• What is Secant?
• Secant (or sec) is the 'flipped version' of the cosine function. If we look at SOH CAH TOA, we see that Cosine is Adjacent over Hypotenuse
 cosine =Adjacent / Hypotenuse
but if we use secant, it would be 'flipped' (called the reciprocal), becoming Hypotenuse over Adjacent
 secant = Hypotenuse / Adjacent

Example
cos(60 degrees) = 0.5 or 1/2
so
sec(60 degrees) = 2 or 2/1
• that was really neat
• "Neat! Whoever named the trig functions must have deeply understood the relationships between them." Then why is the reciprocal function of Sine, Cosecant and Cosine, Secant. Whoever named the trig functions should've made the reciprocal function of sine, secant and the reciprocal function of Cosine, Cosecant.
• Why it can not be Sine@=Csc (90-@)
And
Why it is it Sine@=Cos(90-@)*bold
• 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'.
• i still am confused on the topic...
• Welcome to trig!

In all likelihood, if you can keep at it with decent energy levels, you'll understand better as time goes on.
• how do you find other angles with only the sides given?
• 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.
• What secant cotangent and cosecant about