Trigonometry: Ratios and Identities

• The session improves upon the previous year's trigonometry series, which had 700,000 views.
• The aim is to make the subject more understandable and detailed.
• The current trigonometry session is considered the best yet.
• Content is better "sandwiched," with simpler and more challenging examples added.
• Half of the intermediate questions have been updated to provide new perspectives.
• Updated questions are available in a sheet in the description, with video solutions.
• It's recommended to attempt the sheet after the session.
• The session starts by explaining the chapter's nature and contents.
• Arvind Kalia Sir is the instructor.
• Arvind Kalia is seen as a brother figure who will support students throughout the year; show your support with the hashtag #merabhaimerabhai.
• The speaker aims to simplify everything.
• Direct questions aren't frequently asked in JEE, but the concepts are widely applicable.
• It's an extension of 10th-grade trigonometry but significantly expanded and frequently used, making it important.
• Direct weightage isn't significant, but it's frequently used, making it important.
• In Advanced, direct questions sometimes appear, but application within other problems is common.
• Allied angles, multiple angle formulas, and max/min value of trigonometric expressions should be emphasized.
• A previous 'One Year Road Map' and advice for students whose 11th-grade beginnings were unsteady have already been released.
• Two sessions are available for students whose 11th-grade start was shaky: one offers a month-by-month roadmap for IIT preparation, and the other focuses on regaining track after a shaky start.
• All materials on YouTube are comprehensive, from basic to advanced levels.
• Subscriptions offer additional benefits, currently, there is 25% off subscriptions.

Systems for Measuring Angles

• Sexagesimal and circular are the two main systems for measuring angles.
• Sexagesimal is the traditional degree system (90°, 60°, 45°).
• Mathematicians defined a right angle as a quarter of a circle.
• A degree is 1/90th of a right angle.
• A minute is 1/60th of a degree.
• A second is 1/60th of a minute.
• Circular measures in radians.
• Radians were discovered when a mathematician measured arcs on different-sized circles and determined the ratio of arc length to radius was constant.
• The ratio of arc length (l) to radius (r) defines the angle (θ) in radians: θ = l/r.
• In this system, the angle is a real number.
• 90 degrees equals π/2 radians.
• 180 degrees equals π radians.
• One radian is the angle created when the radius equals the arc length.
• Radians are a number since length and radius are numbers.
• A general formula to convert degrees to radians is x degrees = (π/180) * x radians.
• To convert from radians to degrees, use x radians = (180/π) * x degrees.
• One radian is approximately 57 degrees.
• An unpopular 3rd method is the centesimal system
• 1 Right Angle = 90 Degrees
• 1 Degree = 60 Minutes
• 1 Minutes = 60 Seconds

Radians

• π radians equals approximately 180 degrees.
• π/2 radians equals 90 degrees.
• One radian is in the first quadrant.
• Two radians is in the second quadrant.
• Three radians is in the second quadrant.

Working with Radians and Degrees Together

• Use π = 3.14 for radians, not 180.
• 180 degrees $70
• $ means degree unit
• If a circle has a radius r, the arc length needed for 1 radian is r.
• To convert 10 degrees and 30 minutes into radians:
• Convert 30 minutes to 0.5 degrees.
• Total angle 10.5 degrees.
• Use conversion factor to get radians.
• Radian to degree conversion can be trickier:
• The formula often leaves π in the answer.
• Degrees have subunits to use when calculating.
• Simplification is important.
• One radian is about 57 degrees.
• One minute is equal to 60 seconds.
• Avoid saying "90 degrees" is one right angle, instead use "a quarter of a circle."
• If the length of the Arc was equal to a quarter of a circle with Radius of R, then Theta = 2 * pi * R/ 4 divided by R = pi/2.
• Note as you head further into 11th grade, the use of degrees dies out (Pi focus).
• It is important to know whether Pi means 180 or 3.14; Pi in radian use = 3.14.
• Values get as close as possible with the radian values to see which quadrant they are.
• One Radian angle = First Quandrant
• Two Radians = Second Quadrant
• Three Radians = Third Quadrant
• Five Radians = Fourth Quadrant

Relationship Between Length, Radius, and Theta

• The expression θ = l/r represents an important relationship useful in calculations.

Clock Angles

• The hour hand travels half a degree each minute.
• To find the angle between the minute and hour hand:
• Calculate the angle of the minute hand.
• Calculate the angle of the hour hand.
• Take the difference.

Significance of unit circle

• Key info includes
• circle
• Origin is at center
• Has a 1-unit radius (R)

Cosine and Sine Definitions

• For any angle there is intersection
• (x,y)
• x = cos (Theta)
• y = sin (Theta)
• Degree measurement can be seen visually

Basic Findings from Unit Circle

• You can give any angle and figure out Sine and Cosine.
• Sine and Cosine values are bound between negative one and one.
• Sine and Cosine are defined for Every angle
• Some are pretty
• Some or hard
• Angle measure is X coordinate
• For a vertical line, it is the Y coordinate

Common Angles

• 024 pi : + X axis
• Pi 3pi 5pi : -X axis
• pi/2 5pi/2 9pi/2 4n + pi/2 + Y
• 3pi/27 pi/2 11 pi/2 4n + 3pi/2 - Y

Trig-based Equations

• Degrees is only for the Triangle
• Quadrant only tells you plus or minus

Allied Angle

• Important for chapter
• "I Will Teach you In a way that will put it all in your head!"
• What they are
• Theta and n* pi +- theta
• Are two brothers together
• The direction of the angle are related and tied together

How to work Allied angles

• Key Info:
• Put sign thetha
• Look at Quadrant using All Silver Tea Cups
• Plus or minus Depends on what Quadrant you are in

Fast Moving and Working Tricks

• Sine / Cosines are - at "Super Supplementary" Angles
• Pi + Theta
• Not A FORMAL Notation
• This is only For Easy to Think About and Solve Things

Quick Reminder Table

• Supplement Angle Sign = Same at signs
• Supplement Angle: Cosign = Negative
• Super Supplement Sign or CoSign = Negative
• Radian Measure System
  • pi/6
  • pi/4
  • pi/3
• It easy to move from degree but easy to move to Degree

The quick fix

• A set of angles are tough
• Figure them out like so:
• Use trig property
• Draw triangle
• Sign is based on what is defined: Quadrent
• Example problem from session:
• To Find Angle Between:
• Easy to put the degree in your Triangle
• Look at what cord you are doing and figure it out

Trig Graphs

• Trig graphs are more visual representations
• Important features to remember include:
• the period
• local and absolute Maximums
• Local and absolute minimums.
• The unit Circle

Compound Angle Section

• What requirements must be in place?
• Easy version
• Check to see if the Angle looks Familiar
• Add to make it Easy
• Easy and Quick Math
• Don't forget to remember
• Know what are looking for to quickly math !
• You can know your required values in advance
• Always check Trig tables
• A + B over C = A over C + B /C

Using Other Theorems

• Use: Componendo and Dividendo Theorem
• A relationship trick for solving for equal parameters
• Very important that equal ratios cannot take themselves!
• Good to use as a result.

Multiple Angles

• Remember
• Look at requirement and what is available
• Cut down step
  • Save Time
  • Be efficient
• Remember what you were told
• Square everything and go !
• All trig identities from today are important.
• Also note that power = power and should still give you the proper info

Power Steps

• Square is very important
• A result must stick well
• To many is a time theft "When In doubt follow power !""

Other Key Formulas

• How to go into sign with all !
• With out using crazy method?
• Remember how to right equations
• To know how to make 1, must be able to use one
• SideNote
  • Super Supplementary: Sine and Cosine are Equal

TransFormula Key

• "How are you Doing"
• Don't think they are simple
• They will bite your head if not careful

What do?

• Key Findings
• All are equal and need one

Important Find

• Trig mean to focus on the angle.