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# Trigonometry

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#### Transcript of Trigonometry

Trigonometry Chapter 4 Trig + Unit Circle Radian Measure of central angle subtended in a circle by an arc equal in measure to the radius. 1 Rad = 180/pi

1 Deg = pi/180 Converting drawing in radians Coterminal Angles Arc Length If theta is in radians Arc =radius (theta) Theta +/- 360 n

Theta +/- 2Pi n n must be natural numbers 0,1,2,3....... Infinite numbers of Coterminal Angles Les "formulae" CHAPTER 4.1 x^2 + y^2 = r^2 x^2 + y^2 = 1 Unit Circle Recall:

Pythagorean Theorem radius of 1 difference is the radius of 1 looks like Special

Triangles normally special triangles using unit circle Chapter 4.2 Recall:

Trig Ratios sinX = y/r

cosX = x/r

tanX = y/x sinX = y

cosX = x

tanX = y/x before r = 1, therefore r "cancels" coterminal angles of 45 30 = pi/6

45 = pi/4

60 = pi/3 in radians Reciprocal Ratios the inverse of normal ratios coesecant = 1/sinX

secant = 1/cosX

cotangent = cosX/sinX les ratios don't forget, all

strippers take cash Chapter 4.3 Trig Equations Solve 2(sinX)^2 = 1 [ 0, 2 pi ] 2(sinX)^2 - 1 = 0

(sinX)^2 = 1/2

sinX = +/- 1/rt(2) It can be in all Quadrants it's special ;) YOU'LL NEED IT! x = Pi/4. 3Pi/4, 5Pi/4, 7Pi/4 take in to account

the limits! General form done in radians Theta +/- 2Pi n n is all integers X = Pi/4 + Pi n

x = Pi3/4 + Pi n next step Chapter 5 Graphing Cosine Graph Sine Graph Tangent Graph Transformation 4Cos(2(x-45)) + 1 Range: -3<y<5 Shift = 45 to the right Sinusoïdal Axis = 1 Ones are found at 45*+90n, nEI For sine graph every 1/2 of a period and full period the graph lands on the sinusoidal axis. Every 1/4 period after a sinusoidal intercept is a max or min value. Graph starts on the sinoisoidal axis. Cosine graph starts at max, every period it lands on another max. Asymptotes are found at 90* + 180n, nEI Solving with algebra I.e. 16=6sin(Pi/6)X) + 14 2/6=sin(Pi/6)X) Allow Pi/6 act as X/Theta 1/3= sinX Reference angle = 0.34 X=0.65+ 12n = 5.35+ 12n X= 0.34 and 2.8 Example Example of tangent word question they're like shifted functions of each other :) 9700m Radar Write a fuction relating the angle between the line of sight and the vertical with the horizontal distance of the plane and radar station. Answer: TanX=Y/9700

Y=9700TanX Example for a word question for Cosine/Sin Write a function for a Ferris wheel which rotates every 120 secs. It has a radius of 20m and is 3m above the ground. Ferris wheel starts at bottom. D= (Max + Min)/2

= (43+3)/2

= 23 Amplitude: Max-D= 43-23=20 360 / B = 120

B =3 Therefore: -20cos(3X)+23

and -20sin3(X+30) + 23 applies to a lot of stuff :D These graphs can be transformed! Chapter 6 Trigonometric Identities Pythagorean Identities cos^2x + sin^2x = 1

1 + tan^2x = sec^2x

cot^2x + 1 = csc^2x Other identities cos(A+B) = cosAcosB - sinAsinB

cos(A-B) = cosAcosB + sinAsinB

sin(A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB - cosAsinB

tan(A+B) = (tanA + tanB) / 1 - tanAtanB

tan(A-B) = (tanA - tanB) / 1 + tanAtanB sin2X = 2sinXcosX

cos2X = cos^2X - sin^2X

= 2cos^2X -1

= 1 - 2sin^2X

tan2X = (2tanX)/(1 - tan^2X) addition identities Double-Angle Identities simplify Proof Verify Non permissible values related to the basics a period of cosX and sinx is 2 Pi values that don't work example example Proofs do the equations match? example Trigonometry

Identities and

Equuations an application of everything example it's like a monster chapter of algebra TanX+CscX=0

SinX/CosX+ 1/SinX = 0

Therefore Cosx and SinX cannot = 0

This means X cannt = 0,90,180,360 for a 0<x<360 range Sin^2 x + 6Sinx + 5 (Sinx+1)(SinX+5)

Sinx = -1, SinX = -5

Only Sinx = -1 is possible

therefor only answer is 270* (Tan^2)(Csc^2)(Cos) = Sec

(Sin^2/Cos^2)(1/Sin^2)(Cos)=

Sin^4Cos/(Sin^2)(Cos^2)(Cos)=

1/Cos = Sec

Sec = Sec

Cos195 = Cos(60+135) =CosACosB - SinASinB

= (1/2)(-1/rt2) - (rt3/2)(1/rt2)

= -1 - sqrt3/ 2rt2 Cos^2 + Sin^2 = 1 because in a unit circle if you square the adjacent and opposite you will always get one! The identities are used for most frequently for proofing. The identities can also help you solve an equation.

Full transcript1 Deg = pi/180 Converting drawing in radians Coterminal Angles Arc Length If theta is in radians Arc =radius (theta) Theta +/- 360 n

Theta +/- 2Pi n n must be natural numbers 0,1,2,3....... Infinite numbers of Coterminal Angles Les "formulae" CHAPTER 4.1 x^2 + y^2 = r^2 x^2 + y^2 = 1 Unit Circle Recall:

Pythagorean Theorem radius of 1 difference is the radius of 1 looks like Special

Triangles normally special triangles using unit circle Chapter 4.2 Recall:

Trig Ratios sinX = y/r

cosX = x/r

tanX = y/x sinX = y

cosX = x

tanX = y/x before r = 1, therefore r "cancels" coterminal angles of 45 30 = pi/6

45 = pi/4

60 = pi/3 in radians Reciprocal Ratios the inverse of normal ratios coesecant = 1/sinX

secant = 1/cosX

cotangent = cosX/sinX les ratios don't forget, all

strippers take cash Chapter 4.3 Trig Equations Solve 2(sinX)^2 = 1 [ 0, 2 pi ] 2(sinX)^2 - 1 = 0

(sinX)^2 = 1/2

sinX = +/- 1/rt(2) It can be in all Quadrants it's special ;) YOU'LL NEED IT! x = Pi/4. 3Pi/4, 5Pi/4, 7Pi/4 take in to account

the limits! General form done in radians Theta +/- 2Pi n n is all integers X = Pi/4 + Pi n

x = Pi3/4 + Pi n next step Chapter 5 Graphing Cosine Graph Sine Graph Tangent Graph Transformation 4Cos(2(x-45)) + 1 Range: -3<y<5 Shift = 45 to the right Sinusoïdal Axis = 1 Ones are found at 45*+90n, nEI For sine graph every 1/2 of a period and full period the graph lands on the sinusoidal axis. Every 1/4 period after a sinusoidal intercept is a max or min value. Graph starts on the sinoisoidal axis. Cosine graph starts at max, every period it lands on another max. Asymptotes are found at 90* + 180n, nEI Solving with algebra I.e. 16=6sin(Pi/6)X) + 14 2/6=sin(Pi/6)X) Allow Pi/6 act as X/Theta 1/3= sinX Reference angle = 0.34 X=0.65+ 12n = 5.35+ 12n X= 0.34 and 2.8 Example Example of tangent word question they're like shifted functions of each other :) 9700m Radar Write a fuction relating the angle between the line of sight and the vertical with the horizontal distance of the plane and radar station. Answer: TanX=Y/9700

Y=9700TanX Example for a word question for Cosine/Sin Write a function for a Ferris wheel which rotates every 120 secs. It has a radius of 20m and is 3m above the ground. Ferris wheel starts at bottom. D= (Max + Min)/2

= (43+3)/2

= 23 Amplitude: Max-D= 43-23=20 360 / B = 120

B =3 Therefore: -20cos(3X)+23

and -20sin3(X+30) + 23 applies to a lot of stuff :D These graphs can be transformed! Chapter 6 Trigonometric Identities Pythagorean Identities cos^2x + sin^2x = 1

1 + tan^2x = sec^2x

cot^2x + 1 = csc^2x Other identities cos(A+B) = cosAcosB - sinAsinB

cos(A-B) = cosAcosB + sinAsinB

sin(A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB - cosAsinB

tan(A+B) = (tanA + tanB) / 1 - tanAtanB

tan(A-B) = (tanA - tanB) / 1 + tanAtanB sin2X = 2sinXcosX

cos2X = cos^2X - sin^2X

= 2cos^2X -1

= 1 - 2sin^2X

tan2X = (2tanX)/(1 - tan^2X) addition identities Double-Angle Identities simplify Proof Verify Non permissible values related to the basics a period of cosX and sinx is 2 Pi values that don't work example example Proofs do the equations match? example Trigonometry

Identities and

Equuations an application of everything example it's like a monster chapter of algebra TanX+CscX=0

SinX/CosX+ 1/SinX = 0

Therefore Cosx and SinX cannot = 0

This means X cannt = 0,90,180,360 for a 0<x<360 range Sin^2 x + 6Sinx + 5 (Sinx+1)(SinX+5)

Sinx = -1, SinX = -5

Only Sinx = -1 is possible

therefor only answer is 270* (Tan^2)(Csc^2)(Cos) = Sec

(Sin^2/Cos^2)(1/Sin^2)(Cos)=

Sin^4Cos/(Sin^2)(Cos^2)(Cos)=

1/Cos = Sec

Sec = Sec

Cos195 = Cos(60+135) =CosACosB - SinASinB

= (1/2)(-1/rt2) - (rt3/2)(1/rt2)

= -1 - sqrt3/ 2rt2 Cos^2 + Sin^2 = 1 because in a unit circle if you square the adjacent and opposite you will always get one! The identities are used for most frequently for proofing. The identities can also help you solve an equation.