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# Trigonometric Identities

Trigonometric Identities
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Nie wieder prokastinieren mit unseren Lernerinnerungen. 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.

## What is the main set of trigonometric identities?

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 ${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$.

PROOF:

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:$=\frac{a}{c}×\frac{c}{b}=\frac{a}{b}$

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.

## Worked examples using trigonometric identities

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 $4\left(1-{\mathrm{cos}}^{2}x\right)+8\mathrm{cos}x-7=0$ .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 $y=\mathrm{cos}x$.$4{y}^{2}-8y+3=0\phantom{\rule{0ex}{0ex}}\left(2y-1\right)\left(2y-3\right)=0\phantom{\rule{0ex}{0ex}}y=0.5ory=1.5$Now we need to do x = cos-1(y)We can only perform cos-1(0.5)=60°This is because 1.5 > 1 so we cannot perform a cos-1 function of this.So the only answer is 60°.

Let's look at another example of rearranging trigonometric identities.

Show that the equation $2\mathrm{sin}x=\frac{\left(4\mathrm{cos}x-1\right)}{\mathrm{tan}x}$ 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 $1-{\mathrm{cos}}^{2}x$:$2\left(1-{\mathrm{cos}}^{2}x\right)=4{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}2-2{\mathrm{cos}}^{2}x=4{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}$Now rearrange this equation:$2=6{\mathrm{cos}}^{2}x-\mathrm{cos}x\phantom{\rule{0ex}{0ex}}6{\mathrm{cos}}^{2}x-\mathrm{cos}x-2=0$QED

## What other trigonometric identities can we derive?

Firstly 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.

### Deriving new identities

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.

### Worked examples of new identities

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

$2{\mathrm{tan}}^{2}x+secx=1$to 1 dp.  Graph of y=cosx. Image: Ruben Verhaegh, CC BY-SA 4.0

We can see that if we perform the identity $\mathrm{cos}x=\mathrm{cos}\left(360-x\right)$, 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 $\mathrm{cos}x=\mathrm{cos}\left(360-x\right)$, $x=360$.

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

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

## Trigonometric Identities - Key takeaways

• 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.

## Trigonometric Identities Quiz - Teste dein Wissen

Question

What theorem are Pythagorean identities based on?

Pythagoras Theorem

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