Trigonometric Identities

Trigonometric identities are important to work through a variety of problems and advanced Equations. They allow us to simplify many problems and make situations easier.

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Jetzt kostenlos anmeldenTrigonometric identities are important to work through a variety of problems and advanced Equations. They allow us to simplify many problems and make situations easier.

There are two main formulaic identities that must be learnt to prove and solve other Equations. These are:

and $\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}=\mathrm{tan}x$

Let’s prove these identities starting with ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$.

Firstly let’s draw a triangle with angle θ.

General Triangle of angle θ

Now if we write out expressions for a and b using SOHCAHTOA we get:$a=c\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}b=c\mathrm{cos}\theta $Therefore:

$\frac{a}{c}=\mathrm{sin}\theta \phantom{\rule{0ex}{0ex}}\frac{b}{c}=\mathrm{cos}\theta $Now if we square both of these expressions for sin and cos we get:

$\frac{{a}^{2}}{{c}^{2}}={\mathrm{sin}}^{2}\theta \phantom{\rule{0ex}{0ex}}\frac{{b}^{2}}{{c}^{2}}={\mathrm{cos}}^{2}\theta $Summing these we get:

${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =\frac{{a}^{2}+{b}^{2}}{{c}^{2}}$By Pythagoras' theorem:

${a}^{2}+{b}^{2}={c}^{2}\phantom{\rule{0ex}{0ex}}$Therefore:

$\frac{{a}^{2}+{b}^{2}}{{c}^{2}}=\frac{{c}^{2}}{{c}^{2}}=1\phantom{\rule{0ex}{0ex}}{\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$Now let’s move on to proving $\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{tan}x$. The first half of this Proof is identical to the Proof above.

PROOF:

Firstly let’s draw a triangle with angle θ.

Now if we write out expressions for a and b using SOHCAHTOA we get:So Now if we divide these two expressions for sin and cos:$\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}=\frac{\left(\frac{a}{c}\right)}{\left(\frac{b}{c}\right)}$$=\frac{a}{c}\times \frac{c}{b}=\frac{a}{b}$This is an expression for the opposite side over the adjacent side, therefore:

$\frac{a}{b}=\mathrm{tan}\theta $Therefore:

$\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta}=\mathrm{tan}\theta $Now let’s look at some worked examples where trigonometric identities can be applied.

Solve the equation $4{\mathrm{sin}}^{2}x+8\mathrm{cos}x-7=0$ for $0\le x\le 180.$

SOLUTION:The first thing to do would be to substitute$1-{\mathrm{cos}}^{2}x$for ${\mathrm{sin}}^{2}x$ .The equation now ends up being $4(1-{\mathrm{cos}}^{2}x)+8\mathrm{cos}x-7=0$ .Simplifying this further:$4-4{\mathrm{cos}}^{2}x+8\mathrm{cos}x-7=0$$4{\mathrm{cos}}^{2}x-8\mathrm{cos}x+3=0$Now we can solve this like a quadratic by taking $y=\mathrm{cos}x$.$4{y}^{2}-8y+3=0\phantom{\rule{0ex}{0ex}}(2y-1)(2y-3)=0\phantom{\rule{0ex}{0ex}}y=0.5ory=1.5$Now we need to do x = cosLet's look at another example of rearranging trigonometric identities.

Show that the equation $2\mathrm{sin}x=\frac{(4\mathrm{cos}x-1)}{\mathrm{tan}x}$ can be written as $6{\mathrm{cos}}^{2}x-\mathrm{cos}x-2=0.$

SOLUTION:Firstly let’s rearrange to get rid of any denominators.$2\mathrm{sin}x\mathrm{tan}x=4\mathrm{cos}x-1$Now let’s replace $\mathrm{tan}x$ with $\frac{\mathrm{sin}x}{\mathrm{cos}x}$:$2\mathrm{sin}x\frac{\mathrm{sin}x}{\mathrm{cos}x}=4\mathrm{cos}x-1$$\frac{2{\mathrm{sin}}^{2}x}{\mathrm{cos}x}=4\mathrm{cos}x-1$Now get rid of the denominator by multiplying through by $\mathrm{cos}x$:$2{\mathrm{sin}}^{2}x=4{\mathrm{cos}}^{2}x-\mathrm{cos}x$Now replace ${\mathrm{sin}}^{2}x$ with $1-{\mathrm{cos}}^{2}x$:$2(1-{\mathrm{cos}}^{2}x)=4{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}2-2{\mathrm{cos}}^{2}x=4{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}$Now rearrange this equation:$2=6{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}6{\mathrm{cos}}^{2}x-\mathrm{cos}x-2=0$QEDFirstly we need to know three new bits of terminology:

$secx=\frac{1}{\mathrm{cos}x}\phantom{\rule{0ex}{0ex}}\mathrm{cos}ecx=\frac{1}{\mathrm{sin}x}\phantom{\rule{0ex}{0ex}}cotx=\frac{1}{\mathrm{tan}x}$

These are all reciprocals of standard sin, cos and tan.

Now let’s look at the identity ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$:

If we divide the entire equation by ${\mathrm{cos}}^{2}\left(x\right)$we get:$\frac{{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{cos}}^{2}x}=\frac{1}{{\mathrm{cos}}^{2}x}$Now using the identity $\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{tan}x$:${\mathrm{tan}}^{2}x+1=se{c}^{2}x$This is our first new identity. Now if we divide our entire equation by ${\mathrm{sin}}^{2}x$$\frac{{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{2}x}+\frac{{\mathrm{cos}}^{2}x}{{\mathrm{sin}}^{2}x}=\frac{1}{{\mathrm{sin}}^{2}x}$Now using the identity $\frac{\mathrm{sin}x}{\mathrm{cos}x}=\mathrm{tan}x$, so :$1+\frac{1}{{\mathrm{tan}}^{2}x}=\frac{1}{{\mathrm{sin}}^{2}x}\phantom{\rule{0ex}{0ex}}1+co{t}^{2}x=\mathrm{cos}e{c}^{2}x$Now we have our two new identities:${\mathrm{tan}}^{2}x+1=se{c}^{2}x\phantom{\rule{0ex}{0ex}}co{t}^{2}x+1=\mathrm{cos}e{c}^{2}x$Let’s see them in action in some worked examples.

Solve, for 0 ≤ θ < 360°, the equation:

$2{\mathrm{tan}}^{2}x+secx=1$to 1 dp.We can see that if we perform the identity $\mathrm{cos}x=\mathrm{cos}(360-x)$, the other value of $x$ is $360-131.8=228.2$.

Then we need to perform ${\mathrm{cos}}^{-1}\left(1\right)=0$, again using the identity $\mathrm{cos}x=\mathrm{cos}(360-x)$, $x=360$.

So to 1 decimal place our 4 solutions in degrees are:

$x=131.8,x=228.2,x=0,x=360$

Trigonometric identities are used to derive new formulae and equations.

They can help solve equations involving Trigonometry.

They help us geometrically visualise real-life situations.

They have proofs, which can be adapted from basic Trigonometry.

Images:

Graph of y=cos x: https://commons.wikimedia.org/wiki/File:Cos(x).PNG

sinx/cosx=tanx, sin^2(x)+cos^2(x)=1. 1/cosx=secx

Simply rearrange to the identities listed above and substitute them back in.

Drawing a diagram reveals why each identity works. Regular SOHCAHTOA can show what’s going on.

They can help us solve larger trigonometric equations that cannot be solved otherwise.

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