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## Precalculus

### Course: Precalculus>Unit 2

Lesson 10: Using trigonometric identities

# Trig identity reference

Look up AND understand all your favorite trig identities

## Reciprocal and quotient identities

\sec, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction

\csc, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, sine, left parenthesis, theta, right parenthesis, end fraction

cotangent, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, tangent, left parenthesis, theta, right parenthesis, end fraction

tangent, left parenthesis, theta, right parenthesis, equals, start fraction, sine, left parenthesis, theta, right parenthesis, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction

cotangent, left parenthesis, theta, right parenthesis, equals, start fraction, cosine, left parenthesis, theta, right parenthesis, divided by, sine, left parenthesis, theta, right parenthesis, end fraction

## Pythagorean identities

sine, squared, left parenthesis, theta, right parenthesis, plus, cosine, squared, left parenthesis, theta, right parenthesis, equals, 1, squared
tangent, squared, left parenthesis, theta, right parenthesis, plus, 1, squared, equals, \sec, squared, left parenthesis, theta, right parenthesis
cotangent, squared, left parenthesis, theta, right parenthesis, plus, 1, squared, equals, \csc, squared, left parenthesis, theta, right parenthesis

## Identities that come from sums, differences, multiples, and fractions of angles

These are all closely related, but let's go over each kind.
Angle sum and difference identities
\begin{aligned} \sin(\theta+\phi)&=\sin\theta\cos\phi+\cos\theta\sin\phi\\\\ \sin(\theta-\phi)&=\sin\theta\cos\phi-\cos\theta\sin\phi\\\\ \cos(\theta+\phi)&=\cos\theta\cos\phi-\sin\theta\sin\phi\\\\ \cos(\theta-\phi)&=\cos\theta\cos\phi+\sin\theta\sin\phi \end{aligned}
\begin{aligned} \tan(\theta+\phi)&=\dfrac{\tan\theta+\tan\phi}{1-\tan\theta\tan\phi}\\\\ \tan(\theta-\phi)&=\dfrac{\tan\theta-\tan\phi}{1+\tan\theta\tan\phi} \end{aligned}
Double angle identities
sine, left parenthesis, 2, theta, right parenthesis, equals, 2, sine, theta, cosine, theta
cosine, left parenthesis, 2, theta, right parenthesis, equals, 2, cosine, squared, theta, minus, 1
tangent, left parenthesis, 2, theta, right parenthesis, equals, start fraction, 2, tangent, theta, divided by, 1, minus, tangent, squared, theta, end fraction
Half angle identities
\begin{aligned} \sin\dfrac\theta2&=\pm\sqrt{\dfrac{1-\cos\theta}{2}}\\\\ \cos\dfrac\theta2&=\pm\sqrt{\dfrac{1+\cos\theta}{2}}\\\\ \tan\dfrac{\theta}{2}&=\pm\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}}\\ \\ &=\dfrac{1-\cos\theta}{\sin\theta}\\ \\ &=\dfrac{\sin\theta}{1+\cos\theta}\end{aligned}

## Symmetry and periodicity identities

sine, left parenthesis, minus, theta, right parenthesis, equals, minus, sine, left parenthesis, theta, right parenthesis
cosine, left parenthesis, minus, theta, right parenthesis, equals, plus, cosine, left parenthesis, theta, right parenthesis
tangent, left parenthesis, minus, theta, right parenthesis, equals, minus, tangent, left parenthesis, theta, right parenthesis
\begin{aligned} \sin(\theta+2\pi)&=\sin(\theta)\\\\ \cos(\theta+2\pi)&=\cos(\theta)\\\\ \tan(\theta+\pi)&=\tan(\theta) \end{aligned}

## Cofunction identities

\begin{aligned} \sin\theta&= \cos\left(\dfrac{\pi}{2}-\theta\right)\\\\ \cos\theta&= \sin\left(\dfrac{\pi}{2}-\theta\right)\\\\ \tan\theta&= \cot\left(\dfrac{\pi}{2}-\theta\right)\\\\ \cot\theta&= \tan\left(\dfrac{\pi}{2}-\theta\right)\\\\ \sec\theta&= \csc\left(\dfrac{\pi}{2}-\theta\right)\\\\ \csc\theta&= \sec\left(\dfrac{\pi}{2}-\theta\right) \end{aligned}

## Appendix: All trig ratios in the unit circle

Use the movable point to see how the lengths of the ratios change according to the angle.

## Want to join the conversation?

• why do ppl invent triangles and calculus T.T
• People don't invent mathematics, they discover it.
• I am kind of struggling on solving sinusoidal equations (advanced) since I don't do all the identities. I don't check all of the solutions.
Here is some that I know:
sin(θ)=(θ+360)
sin(θ)=pi-θ
sin(θ)=θ+2pi
cos(θ)=2pi-θ
cos(θ)=θ+2pi
is there any others missing? am I doing anything wrong?
• First of all, we should probably make the notation a bit more rigorous, because the way you've phrased it isn't quite correct. Instead, write:
sin(θ)=sin(θ+360)=sin(θ+2pi)
sin(θ)=sin(pi-θ)
sin(θ)=sin(θ+2pi) see above
cos(θ)=cos(2pi-θ)
cos(θ)=cos(θ+2pi)
... and yes, there are lots of others - technically, an infinite number of others since sin and cos are periodic and repeat every 2pi, positive or negative. So, for example, sin(θ)=sin(θ+2npi), where n is any integer.

I'd probably add to the list:
sin(-θ)=-sin(θ)
cos(-θ)=cos(θ),
if we're sticking to sin(a)=sin(b) and cos(a)=cos(b) style identities.
• How do you prove the half-angle identities?
• The easiest way is to see that cos 2φ = cos²φ - sin²φ = 2 cos²φ - 1 or 1 - 2sin²φ by the cosine double angle formula and the Pythagorean identity. Now substitute 2φ = θ into those last two equations and solve for sin θ/2 and cos θ/2. Then the tangent identity just follow from those two and the quotient identity for tangent.
• I have a table of trig identities in my Calculus textbook that has the double cosine identity as:
cos 2x = cos^2 x - sin^2 x
Makes sense, because that's the way you would get it if you applied the rule of adding 2 different angles. How do you get from there to what they have here:
cos 2x = 2cos^2 x - 1?
• The Pythagorean identity states:
sin²𝑥 + cos²𝑥 = 1
This means:
sin²𝑥 = 1 - cos²𝑥
The standard double cosine identity is:
cos 2𝑥 = cos²𝑥 - sin²𝑥
Substituting for sin²𝑥:
cos 2𝑥 = cos²𝑥 - (1 - cos²𝑥)
cos 2𝑥 = 2cos²𝑥 - 1
Comment if you have questions.
• okay this article is great... but i really wish i Had seen it before some of the exercises that came before it. i had to puzzle a lot of those out and it took me longer than it would have had i seen this article. it seems (at least to me) that its a little out of place. vote if you agree!
• While they tend place the articles last, I've learned it's always best to read them first! (Or at least before the exercises.)
• In the Angle sum explanation diagram, why is the bottom triangle's adjacent side to angle theta (cos theta)(cos phi)?
• The bottom triangle is a right triangle with hypotenuse length h = cos phi. So if x were your unknown side, doing normal trig on it gives cos theta = x/h = x / (cos phi), or in other words x = (cos theta)(cos phi). All of the sides in that diagram are defined in the same way, relative to the one side that was defined to be of length 1.
• I still don't know how to get the half angle identities.Who can help me?
• I won't do all three, but you can get the idea from the cosine case.
I assume you're comfortable with the double angle formula: cos(2θ) = cos²θ - sin²θ

Make the substitution φ = 2θ or θ = φ/2

cos(φ) = cos²(φ/2) - sin²(φ/2)
Using the Pythagorean identity we get
cos(φ) = cos²(φ/2) - (1 - cos²(φ/2))
= 2cos²(φ/2) -1

Solving for cos(φ/2) gives
cos²(φ/2) = (cos(φ) + 1)/2
or
cos(φ/2) = ±√((cos(φ) + 1)/2)
Which is the result we wanted.

Now once you have that, you can get the sine case by substituting for sin(φ/2) in terms of cosines

ie √(1 - sin²(φ/2)) = √((cos(φ) + 1)/2)

or (1 - sin²(φ/2)) = (cos(φ) + 1)/2
or sin²(φ/2) = 1 - (cos(φ) + 1)/2

or sin²(φ/2) = 2/2 - (cos(φ) + 1)/2
= (1 - cos(φ))/2

Hence sin(φ/2) = ±√((1 - cos(φ))/2)
Note that the half angle formula for sine gives a result in terms of cosine.

I'll leave the case of tan(φ/2) to you.
• My teacher, as well as textbook, say that the cosine symmetry identity is cos(−θ)=-cos(θ)
but this says that it's cos(−θ)=+cos(θ) which one is correct?
• The correct identity is:
cos(−θ)=+cos(θ)

Your teacher and the book probably mixed up sine and cosine. The sine identity is:
sin(−θ)=-sin(θ)
• Do you need to remember all these identities or at least know how to derive them using the unit circle.
• It's okay just know how to derive them using the unit circle but if you remember all of them, it'll be faster when you solve questions. And I recommend you to remember it because when you are taking a test, you don't have time to derive using the unit circle.
+ It's not that hard to remember though. There is a pattern. And once you proved why it is then it'll be way easier to memorize it.