trig

[ydoaPs]

since i began posting on this site, i have seen more than one thread like "what is sin, cos, ...?". so, somewhat inspired by daves two calculus lessons, decided to post the relevent parts of my trig notes. it was a one semester class, so it will probably be really basic. note: i will not cover graphing and inverse trig function.

 

i will begin by introducing a unit of measurement of angles, because i find them much easier to work with. that unit is radian. the name will make sense after the description. we have a circle whose center is at the origin. as we all know, the circumference of a circle is [math]C=2{\pi}r[/math]. assume the radius of the circle is one. the circumference can be thought of as the full rotation of the radius, so a full rotation is [math]2{\pi}=360^0[/math]. half a rotation is [math]\pi=180^0[/math]. a forth of a rotation is [math]\frac{\pi}{2}=90^0[/math] and so on. angles are measured from the positive x-axis(initial side) in a counter clockwise manner to the terminal side. negative angles are clockwise. to convert radians to degrees, multiply the radian measurement by [math]\frac{180}{\pi}[/math]. to convert from degrees to radians multpily the degree measurement by [math]\frac{pi}{180}[/math]

 

every angle has a reference angle([math]\alpha\angle[/math]. a reference angle is the smallest positive acute angle made by the terminal side of [math]\theta[/math] and the x-axis. in the first quadrant, [math]\alpha\angle=\theta[/math]. in the second quadrant, [math]\alpha\angle=\pi-\theta[/math]. in the third, [math]\alpha\angle=\theta-\pi[/math]. in the fourth, [math]\alpha\angle=2\pi-\theta[/math]. trig functions of [math]\theta=\underline{+}same function of \alpha\angle[/math]

 

each angle also has an infinite number of coterminal angles. coterminal angles are angles that have the same terminal side(kinda makes sense, huh).coterminal angle=[math]\theta\underline{+}n2\pi[/math]

 

the trig functions: sin, cos, tan, csc, sec, cot are all ratios of the sides of a right triangle. each angle has a specific value for each of the trig functions.

sin and cos, sec and csc, tan and cot are what are called cofunctions. cofunctions are positive in the same quadrant. in the first quadrant, all functions are positive. in the second, sin and csc are positive. in the third, tan and cot are positive. in the fourth, cos and sec are positive. the trig function of any acute angle equals the cofunction of said angle's complement.

 

sin and csc, cos and sec, tan and cot are reciprocal functions. that will make sense once you see their definitions and identities

 

[math]sin=\frac{opposite side}{hypotenuse}[/math]

[math]cos=\frac{adjacent side}{hypotenuse}[/math]

[math]tan=\frac{opposite side}{adjacent side}[/math]

[math]csc=\frac{hypotenuse}{opposite side}[/math]

[math]sec=\frac{hypotenuse}{adjacent side}[/math]

[math]cot=\frac{adjacent side}{opposite side}[/math]

 

reciprocal identities

[math]sin\theta=\frac{1}{scs\theta}[/math]

[math]csc\theta=\frac{1}{sin\theta}[/math]

[math]cos\theta=\frac{1}{sec\theta}[/math]

[math]sec\theta=\frac{1}{cos\theta}[/math]

[math]tan\theta=\frac{1}{cot\theta}[/math]

[math]cot\theta=\frac{1}{tan\theta}[/math]

 

ratio identites

[math]tan\theta=\frac{sin\theta}{cos\theta}[/math]

[math]cot\theta=\frac{cos\theta}{sin\theta}[/math]

 

pythagorean identities

[math]sin^2\theta+cos^2\theta=1[/math]

[math]1+tan^2\theta=sec^2\theta[/math]

[math]1+cot^2\theta=sec^2\theta[/math]

 

cofunction identities

[math]sin(\frac{\pi}{2}-\theta)=cos\theta[/math]

[math]cos(\frac{\pi}{2}-\theta)=sin\theta[/math]

[math]cos(\frac{\pi}{2}-\theta)=sin\theta[/math]

[math]tan(\frac{\pi}{2}-\theta)=cot\theta[/math]

[math]cot(\frac{\pi}{2}-\theta)=tan\theta[/math]

[math]sec(\frac{\pi}{2}-\theta)=csc\theta[/math]

[math]scs(\frac{\pi}{2}-\theta)=sec\theta[/math]

 

even/odd identities

[math]sin(-\theta)=-sin\theta[/math]

[math]cos(-\theta)=cos\theta[/math]

[math]tan(-\theta)=-tan\theta[/math]

[math]csc(-\theta)=-csc\theta[/math]

[math]sec(-\theta)=sec\theta[/math]

[math]cot(-\theta)=-cot\theta[/math]

 

solving triangles(side a is opposite anlge alpha; side b is opposite angle beta; side c is opposite angle gamma)

law of sines-[math]\frac{a}{sin\alpha\frac{b}{sin\beta}=\frac{c}{sin{\gamma}}[/math]

law of cosines-[math]c^2=a^2+b^2-2abcos\gamma[/math] *note: when gamma is a right angle, law of cosines turns into pythagorean theorem*

 

area of triangles

[math]A=\frac{1}{2}absin\gamma[/math]

[math]A=\sqrt{s(s-a)(s-b)(s-c)}[/math], when [math]s=\frac{a+b+c}{2}[/math]

 

now that you have all of that, here are some formulas

 

double angle formulas

[math]sin2\theta=2sin\thetacos\theta[/math]

[math]cos2\theta=cos^2\theta-sin^2\theta[/math]

[math]cos2\theta=1-2sin^2\theta[/math]

[math]cos2\theta=2cos^2\theta-1[/math]

[math]tan2\theta=\frac{2tan\theta}[1-tan^2\theta}[/math]

 

half angle formulas

[math]sin\frac{\theta}{2}=\sqrt{\frac{1-cos\theta}{2}}[/math]

[math]cos\frac{\theta}{2}=\sqrt{\frac{1+cos\theta}{2}}[/math]

[math]tan\frac{\theta}{2}=\sqrt{\frac{1-cos\theta}{1+cos\theta}}[/math]

[math]tan\frac{\theta}{2}=\frac{1-cos\theta}{sin\theta}[/math]

[math]tan\frac{\theta}{2}=\frac{sin\theta}{1+cos\theta}[/math]

 

power reducing formulas

[math]sin^2\theta=\frac{1-cos2\theta}{2}[/math]

[math]cos^2\theta=\frac{1+cos2\theta}{2}[/math]

[math]tan^2\theta=\frac{1-cos2\theta}{1+cos2\theta}[/math]

 

sum and difference formulas

[math]sin(\alpha+\beta)=sin\alphacos\beta+cos\alphasin\beta[/math]

[math]sin(\alpha-\beta)=sin\alphacos\beta-cos\alphasin\beta[/math]

[math]cos(\alpha+\beta)=cos\alphacos\beta-sin\alphasin\beta[/math]

[math]cos(\alpha-\beta)=cos\alphacos\beta+sin\alphasin\beta[/math]

[math]tan(\alpha+\beta)=\frac{tan\alpha+tan\beta}{1-tan\alphatan\beta}[/math]

[math]tan(\alpha-\beta)=\frac{tan\alpha-tan\beta}{1+tan\alphatan\beta}[/math]

 

sum and difference to product formulas

[math]sin\alpha+sin\beta=2sin\frac{1}{2}(\alpha+\beta)cos\frac{1}{2}(\alpha-\beta)[/math]

[math]sin\alpha-sin\beta=2cos\frac{1}{2}(\alpha+\beta)sin\frac{1}{2}(\alpha-\beta)[/math]

[math]cos\alpha+cos\beta=2cos\frac{1}{2}(\alpha+\beta)cos\frac{1}{2}(\alpha-\beta)[/math]

[math]cos\alpha-cos\beta=-2sin\frac{1}{2}(\alpha+\beta)sin\frac{1}{2}(\alpha-\beta)[/math]

 

product to sum and difference formulas

[math]sin\alphasin\beta=\frac{1}{2}[cos(\alpha-\beta)+cos(\alpha+\beta)][/math]

[math]cos\alphacos\beta=\frac{1}{2}[cos(\alpha-\beta)+cos(\alpha+\beta)][/math]

[math]sin\alphacos\beta=\frac{1}{2}[sin(\alpha+\beta)+sin(\alpha-\beta)][/math]

 

if you have a calculator, then you don't really need the formula's, but if you don't i'm gonna post a trig chart.

[Dave]

That's a pretty conclusive list of things Unfortunately the latex isn't working (for some reason) so I haven't had time to check it through properly. Thanks

[ydoaPs]

feel free to add anything that you don't see that you think should be here. i also didn't put solving equations because that is fairly intuitive.