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

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.

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.

What secant cotangent and cosecant about

Adding on to ❄RAanchal❄, secant, cosecant, and cotangent are the reciprocal functions of cosine, sine, and tangent. They are not their inverse functions. The inverse functions are arccos, arcsin, and arctan.

So does sin(θ)=cos(90∘ + theta)?

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.

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 ```

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'.

Main content

Trigonometry

Course: Trigonometry > Unit 1

Lesson 7: The reciprocal trigonometric ratios

Sine & cosine of complementary angles

Google Classroom

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

[I'm skeptical. Please show me an example.]

Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90degrees.

[Help! Please break this down for me.]

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 cosine	sine, 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 cotangent	tangent, 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.

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Mike Mahaney
Posted 8 years ago. Direct link to Mike Mahaney's post “how can you study more ef...”
how can you study more effectively to do better on tests?
Button navigates to signup pageComment on Mike Mahaney's post “how can you study more ef...”
(6 votes)
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- Michelle Davidson
  Posted 8 years ago. Direct link to Michelle Davidson's post “I'd say just keep practic...”
  I'd say just keep practicing problems and remember Soh Cah Toa. I'd also memorize the two triangles.
  Comment on Michelle Davidson's post “I'd say just keep practic...”
  (54 votes)
SC
Posted 8 years ago. Direct link to SC's post “So, what kind of question...”
So, what kind of questions are related to these confunctions? Im kind of confused because i don't know when to use this
Button navigates to signup pageComment on SC's post “So, what kind of question...”
(6 votes)
Answer
- Jayatra Saxena
  Posted 8 years ago. Direct link to Jayatra Saxena's post “look sarah when you have ...”
  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
  Comment on Jayatra Saxena's post “look sarah when you have ...”
  (23 votes)
thejasonwu8
Posted 6 years ago. Direct link to thejasonwu8's post “What is Secant?”
What is Secant?
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(2 votes)
Answer
- Justin
  Posted 6 years ago. Direct link to Justin's post “Secant (or sec) is the 'f...”
  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
  Button navigates to signup page
  (21 votes)
shafikulislam215
Posted 2 months ago. Direct link to shafikulislam215's post “that was really neat”
that was really neat
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(11 votes)
Answer
Emraan
Posted 8 years ago. Direct link to Emraan's post “Why it can not be _Sine@=...”
Why it can not be Sine@=Csc (90-@)
And
Why it is it Sine@=Cos(90-@)*bold
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(2 votes)
Answer
- VikingTheDude
  Posted 7 years ago. Direct link to VikingTheDude's post “`sin` stands for _sine_. ...”
  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'.
  Button navigates to signup page
  (13 votes)
𝔸𝕤𝕥𝕒𝕣𝔾𝕒𝕝𝕒𝕩𝕚𝕖𝕤
Posted 3 years ago. Direct link to 𝔸𝕤𝕥𝕒𝕣𝔾𝕒𝕝𝕒𝕩𝕚𝕖𝕤's post “i still am confused on th...”
i still am confused on the topic...
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(3 votes)
Answer
- Alexis DRAI
  Posted 3 years ago. Direct link to Alexis DRAI's post “Welcome to trig! In all ...”
  Welcome to trig!
  
  In all likelihood, if you can keep at it with decent energy levels, you'll understand better as time goes on.
  Comment on Alexis DRAI's post “Welcome to trig! In all ...”
  (9 votes)
asmaa adam
Posted 3 years ago. Direct link to asmaa adam's post “how do you find other ang...”
how do you find other angles with only the sides given?
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(4 votes)
Answer
- Hecretary Bird
  Posted 3 years ago. Direct link to Hecretary Bird's post “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.
  Button navigates to signup page
  (5 votes)
philipoheneatao
Posted 3 years ago. Direct link to philipoheneatao's post “What secant cotangent and...”
What secant cotangent and cosecant about
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(3 votes)
Answer
- obiwan kenobi
  Posted 3 years ago. Direct link to obiwan kenobi's post “Adding on to ❄RAanchal❄, ...”
  Adding on to ❄RAanchal❄, secant, cosecant, and cotangent are the reciprocal functions of cosine, sine, and tangent. They are not their inverse functions. The inverse functions are arccos, arcsin, and arctan.
  Button navigates to signup page
  (5 votes)
Sai Maddimsetty
Posted 9 days ago. Direct link to Sai Maddimsetty's post “"Neat! Whoever named the ...”
"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.
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aq9818
Posted 3 years ago. Direct link to aq9818's post “So does sin(θ)=cos(90∘ + ...”
So does sin(θ)=cos(90∘ + theta)?
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(3 votes)
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- Jerry Nilsson
  Posted 3 years ago. Direct link to Jerry Nilsson's post “sin(𝜃) = cos(90° + 𝜃) i...”
  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.
  Button navigates to signup page
  (5 votes)