Difference between revisions of "Trigonometric identities"
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Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at <math> \sin(\alpha+(-\beta))</math> and we can derive the sine angle subtraction formula using the sine angle addition formula. | Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at <math> \sin(\alpha+(-\beta))</math> and we can derive the sine angle subtraction formula using the sine angle addition formula. | ||
− | + | *<math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> || <math>\sin(\alpha - \beta) = \sin \alpha \cos \beta - \sin \beta \cos \alpha</math> | |
− | + | *<math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> || <math>\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta</math> | |
− | + | *<math>\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta} </math> || <math>\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta} </math> | |
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We can prove <math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> easily by using <math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> and <math>\sin(x)=\cos(90-x)</math>. | We can prove <math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> easily by using <math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> and <math>\sin(x)=\cos(90-x)</math>. | ||
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Double angle identities are easily derived from the angle addition formulas by just letting <math> \alpha = \beta </math>. Doing so yields: | Double angle identities are easily derived from the angle addition formulas by just letting <math> \alpha = \beta </math>. Doing so yields: | ||
− | + | <math>\sin 2\alpha = </math>2\sin \alpha \cos \alpha<math> | |
− | + | </math>\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha<math> | |
− | + | </math>=2\cos^2 \alpha - 1<math> | |
− | + | </math>=1-2\sin^2 \alpha<math> | |
− | + | </math>\tan 2\alpha <math> || = || </math>\frac{2\tan \alpha}{1-\tan^2\alpha} <math> | |
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== Half Angle Identities == | == Half Angle Identities == | ||
− | Using the double angle identities, we can now derive half angle identities. The double angle formula for cosine tells us <math>\cos 2\alpha = 2\cos^2 \alpha - 1 < | + | Using the double angle identities, we can now derive half angle identities. The double angle formula for cosine tells us </math>\cos 2\alpha = 2\cos^2 \alpha - 1 <math>. Solving for </math>\cos \alpha <math> we get </math>\cos \alpha =\pm \sqrt{\frac{1 + \cos 2\alpha}2}<math> where we look at the quadrant of </math>\alpha <math> to decide if it's positive or negative. Likewise, we can use the fact that </math>\cos 2\alpha = 1 - 2\sin^2 \alpha <math> to find a half angle identity for sine. Then, to find a half angle identity for tangent, we just use the fact that </math>\tan \frac x2 =\frac{\sin \frac x2}{\cos \frac x2} <math> and plug in the half angle identities for sine and cosine. |
To summarize: | To summarize: | ||
− | + | </math> \sin \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}2} <math> | |
− | + | </math> \cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2} <math> | |
− | + | </math> \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} <math> | |
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== Even-Odd Identities == | == Even-Odd Identities == | ||
− | <math>\sin (-\theta) = -\sin (\theta) < | + | </math>\sin (-\theta) = -\sin (\theta) <math> |
− | <math>\cos (-\theta) = \cos (\theta) < | + | </math>\cos (-\theta) = \cos (\theta) <math> |
− | <math>\tan (-\theta) = -\tan (\theta) < | + | </math>\tan (-\theta) = -\tan (\theta) <math> |
− | <math>\csc (-\theta) = -\csc (\theta) < | + | </math>\csc (-\theta) = -\csc (\theta) <math> |
− | <math>\sec (-\theta) = \sec (\theta) < | + | </math>\sec (-\theta) = \sec (\theta) <math> |
− | <math>\cot (-\theta) = -\cot (\theta) < | + | </math>\cot (-\theta) = -\cot (\theta) <math> |
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(Otherwise known as sum-to-product identities) | (Otherwise known as sum-to-product identities) | ||
− | * <math>\sin \theta \pm \sin \gamma = 2 \sin \frac{\theta\pm \gamma}2 \cos \frac{\theta\mp \gamma}2< | + | * </math>\sin \theta \pm \sin \gamma = 2 \sin \frac{\theta\pm \gamma}2 \cos \frac{\theta\mp \gamma}2<math> |
− | * <math>\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2< | + | * </math>\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2<math> |
− | * <math>\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2< | + | * </math>\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2<math> |
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The extended [[Law of Sines]] states | The extended [[Law of Sines]] states | ||
− | *<math>\frac a{\sin A} = \frac b{\sin B} = \frac c{\sin C} = 2R.< | + | *</math>\frac a{\sin A} = \frac b{\sin B} = \frac c{\sin C} = 2R.<math> |
== Law of Cosines == | == Law of Cosines == | ||
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The [[Law of Cosines]] states | The [[Law of Cosines]] states | ||
− | *<math>a^2 = b^2 + c^2 - 2bc\cos A. < | + | *</math>a^2 = b^2 + c^2 - 2bc\cos A. <math> |
== Law of Tangents == | == Law of Tangents == | ||
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The [[Law of Tangents]] states | The [[Law of Tangents]] states | ||
− | *<math>\frac{b - c}{b + c} = \frac{\tan\frac 12(B-C)}{\tan \frac 12(B+C)}.< | + | *</math>\frac{b - c}{b + c} = \frac{\tan\frac 12(B-C)}{\tan \frac 12(B+C)}.<math> |
== Other Identities = | == Other Identities = | ||
− | *<math>|1-e^{i\theta}|=2\sin\frac{\theta}{2} | + | *</math>|1-e^{i\theta}|=2\sin\frac{\theta}{2}$ |
==See also== | ==See also== |
Revision as of 18:16, 25 October 2007
Trigonometric identities are used to manipulate trig equations in certain ways. Here is a list of them:
Contents
Basic Definitions
The six basic trigonometric functions can be defined using a right triangle:
![Righttriangle.png](https://wiki-images.artofproblemsolving.com//0/00/Righttriangle.png)
The six trig functions are sine, cosine, tangent, cosecant, secant, and cotangent. They are abbreviated by using the first three letters of their name (except for cosecant which uses ). They are defined as follows:
Reciprocal Relations
From the last section, it is easy to see that the following hold:
Another useful identity that isn't a reciprocal relation is that .
Pythagorean Identities
Using the Pythagorean Theorem on our triangle above, we know that . If we divide by
we get
which is just
. Dividing by
or
instead produces two other similar identities. The Pythagorean Identities are listed below:
(Note that the second two are easily derived by dividing the first by and
)
Angle Addition/Subtraction Identities
Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at and we can derive the sine angle subtraction formula using the sine angle addition formula.
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We can prove easily by using
and
.
Double Angle Identities
Double angle identities are easily derived from the angle addition formulas by just letting . Doing so yields:
2\sin \alpha \cos \alpha$$ (Error compiling LaTeX. Unknown error_msg)\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha$$ (Error compiling LaTeX. Unknown error_msg)=2\cos^2 \alpha - 1$$ (Error compiling LaTeX. Unknown error_msg)=1-2\sin^2 \alpha$$ (Error compiling LaTeX. Unknown error_msg)\tan 2\alpha
\frac{2\tan \alpha}{1-\tan^2\alpha}
\cos 2\alpha = 2\cos^2 \alpha - 1
\cos \alpha
\cos \alpha =\pm \sqrt{\frac{1 + \cos 2\alpha}2}
\alpha
\cos 2\alpha = 1 - 2\sin^2 \alpha
\tan \frac x2 =\frac{\sin \frac x2}{\cos \frac x2} $and plug in the half angle identities for sine and cosine.
To summarize:$ (Error compiling LaTeX. Unknown error_msg) \sin \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}2} $$ (Error compiling LaTeX. Unknown error_msg) \cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2} $$ (Error compiling LaTeX. Unknown error_msg) \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} \sin (-\theta) = -\sin (\theta) $$ (Error compiling LaTeX. Unknown error_msg)\cos (-\theta) = \cos (\theta) $$ (Error compiling LaTeX. Unknown error_msg)\tan (-\theta) = -\tan (\theta) $$ (Error compiling LaTeX. Unknown error_msg)\csc (-\theta) = -\csc (\theta) $$ (Error compiling LaTeX. Unknown error_msg)\sec (-\theta) = \sec (\theta) $$ (Error compiling LaTeX. Unknown error_msg)\cot (-\theta) = -\cot (\theta) $==Prosthaphaeresis Identities==
(Otherwise known as sum-to-product identities)
- $ (Error compiling LaTeX. Unknown error_msg)\sin \theta \pm \sin \gamma = 2 \sin \frac{\theta\pm \gamma}2 \cos \frac{\theta\mp \gamma}2
\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2
\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2$== Law of Sines ==
{{main|Law of Sines}} The extended [[Law of Sines]] states
- $ (Error compiling LaTeX. Unknown error_msg)\frac a{\sin A} = \frac b{\sin B} = \frac c{\sin C} = 2R.$== Law of Cosines ==
{{main|Law of Cosines}} The [[Law of Cosines]] states
- $ (Error compiling LaTeX. Unknown error_msg)a^2 = b^2 + c^2 - 2bc\cos A. $== Law of Tangents ==
{{main|Law of Tangents}} The [[Law of Tangents]] states
- $ (Error compiling LaTeX. Unknown error_msg)\frac{b - c}{b + c} = \frac{\tan\frac 12(B-C)}{\tan \frac 12(B+C)}.
|1-e^{i\theta}|=2\sin\frac{\theta}{2}$