10th Class Math: Trigonometry

• The session is a Class 10th math marathon for gap days, focusing on exam preparation
• The goal is to score 100/100 in the math exam.
• The marathon consists of stage 1 with Trigonometry on March 1st, followed by Real Numbers and Polynomials.
• Stage 2 includes polishing chapters from March 7th to 9th, and final revisions on March 10th are scheduled.
• After each class, a "GDP Square" i.e Gap Days Practice Paper will be provided, with discussions the next morning.

Trigonometry Introduction

• Trigonometry involves measuring three sides, mainly dealing with triangles, and more specifically, right-angled triangles.
• The right-angled triangle has three components: hypotenuse, base, and perpendicular.
• The Hypotenuse is "always" opposite the 90-degree angle.
• How to determine the Base and Perpendicular: depends on the "reference angle"
• The side opposite the reference angle (Theta "Θ") is the perpendicular, while the adjacent side is the base.

Trigonometric Ratios

• Trigonometric ratios are derived by dividing the side lengths
• Key ratios: sine, cosine, tangent, cosecant, secant, and cotangent.
• Sine of angle Theta (sin Θ) = Perpendicular / Hypotenuse (P/H).
• Cosine of angle Theta (cos Θ) = Base / Hypotenuse (B/H).
• Tangent of angle Theta (tan Θ) = Perpendicular / Base (P/B).
• Cosecant of angle Theta (cosec Θ) = Hypotenuse / Perpendicular (H/P).
• Secant of angle Theta (sec Θ) = Hypotenuse / Base (H/B).
• Cotangent of angle Theta (cot Θ) = Base / Perpendicular (B/P).

Mnemonic

• Mnemonic to memorize the ratios: "Papa Bidi Pioge? Haan Haan Beta".

Reciprocal relationships

• Sine and cosecant are reciprocal to each other.
• Cosine and secant are reciprocal to each other.
• Tangent and cotangent are reciprocal to each other.
• Sin (Θ) = 1/cosec (Θ).
• Cos (Θ) = 1/sec (Θ).
• Tan (Θ) = 1/cot (Θ).
• Tan (Θ) = Sin (Θ) / Cos (Θ).
• Cot (Θ) = Cos (Θ) / Sin (Θ).

Solving Trigonometric Problems

• If Sin A = 3/5, find Cos A and Cot A:
• Sin A = P/H = 3/5 (Given).
• To find the unknown side, the Pythagorean theorem is used i.e Hypotenuse² = Perpendicular² + Base².
• A constant "k" is used with ratios: P = 3k, H = 5k etc.
• The value of k often cancels out during calculation.
• Cos A is then calculated as Adjacent/Hypotenuse.
• Cot A is calculated as Adjacent/Opposite.
• It may show up as expressions as well

Expressing Trigonometric ratios

• To express one trigonometric ratio in terms of another, e.g., sec (Θ) in terms of cot (Θ):
• Give the trigonometric ratio a variable i.e let cot (Θ) = x.
• Use trigonometric identities and relationships to express the required ratio in terms of x, then substitute back.
• If sec(Θ) is about expressing in terms of cot (Θ), then sec (Θ) = √cot²(Θ) + 1 / cot (Θ).

Trigonometric Values of Standard Angles

• Common angles include 0°, 30°, 45°, 60°, and 90°.
• Important Trigonometric values:
• Sin 0° = 0, Sin 30° = 1/2, Sin 45° = 1/√2, Sin 60° = √3/2, Sin 90° = 1
• Cos 0° = 1, Cos 30° = √3/2, Cos 45° = 1/√2, Cos 60° = 1/2, Cos 90° = 0
• Tan 0° = 0, Tan 30° = 1/√3, Tan 45° = 1, Tan 60° = √3, Tan 90° = Not defined
• A trick is "Sign and Tan" must be known.

Application example

• Example value equation: value of Cos 60 - Sin 45 is determined based on substituting the known values of each term.

Solving for angles

• If sin (Θ) = 1/2, use the values table to identify the angle i.e. 30 (Θ). That is therefore Sin (30°) = 1/2.

OZI Formulas

• Sin² (Θ) + Cos² (Θ) = 1
• Sec² (Θ) - Tan² (Θ) = 1
• Cosec² (Θ) - Cot² (Θ) = 1

Modification examples

• Sec² (Θ) = 1 + Tan² (Θ)
• Tan² (Θ) = Sec² (Θ) - 1
• Cosec² (Θ) = 1 + Cot² (Θ)
• Cot² (Θ) = Cosec² (Θ) - 1
• Sin² (Θ) = 1 - Cos² (Θ)
• Cos² (Θ) = 1 - Sin² (Θ)

Application examples

• Show 9Sec²A - 9 Tan²A. Answer = 9.

Steps

• Take 9 common
• Formula of OZI is = 1 Hense proved = 9

Proving Equations

• Proof needs L.H.S = R.H.S Steps when the equations involve tangent
• Turn Tan and Cot
• Add / subtract to cancel them out

Things to look out for

• Write important formulas out for memory sake

Assertions: Reciprocals and OZI Formulas

Key take away: Reciprocal + OZI formulas will determine your next steps

• The takeaway is that you learn how to interrelate information with another, as shown with OZI and with sin cos tan relationship

Reciprocal example

If: Sec Θ + Tan Θ = p Then to remember : Sec Θ - Tan Θ = 1/p (With practice)

Remember!

When one thing is seen, assume a given with its relationship (As shown with previous note) Apply: Sec Θ and Tan Θ and a given asks a value for OZI side

1. Sec Θ + Tan Θ is given, immediately determine Sec Θ - Tan Θ = 1/p = this is value

Tips and tricks

When comes to section B, solve side by side for quick comparison

Quick equation tip to remember!

See OZI side, change OZI = opposite fraction = answer to reciprocal side.

Equation Example

If sec θ + tan θ = p; find the value of tan θ.

• Answer*: p² - 1 / 2p

Remember

If see these keywods 1.CoSec - Cot : You can figure out side from each other : 2.Then to find, use to put variables that then can be easy to interlink or put into easy formula.

Quick Key points to remember the concept

When one side is there 1.Figure out that one side that is opposite by memorization 2.From here if this is sec or tan, remember OZI or other relationship between values.

Word association to remember Key concepts

Example 1: Square, OZI formula = relate!

Key equation remember.

• Find a L term, try to relate something or an idea or a formula you know = create!

Side notes when doing this type

Remember always show working out for every type

Key thing in trig

If all else fails, follow the chain of logic with common sense in the OZI FORMULAS

• This may also help with solving the equation backwards.

One way to remeber equation

• Try to see how you are doing a certain step Ask = what are we applying at each stage.

Remember what needs to be answered or solved!

• LHS RHS needs to be answered.

Chain of thought

• Try to get some value or form that we can chain later, with formula that can be expanded such is squaring
• Then relate and simplify.

Real Numbers: Chapter 2 Summary

What Chapter is abou

• Topic focuses on two main concepts
  1. HCF/LCM
  2. Irrationality proof Key takeaway: This part HCF/LCM part will contain one McQ while second part will contain long ones, most likely one or two in it.

Tips in real numbers

Write all side notes that can be remembered

How to solve for HCF LCM: Factors matter = to HCF and LCM

• Get common then extract into a result
• LCM extract in max power possible. The above shows how to factor the HCF and LCM from a set of terms = easy way to solve LCM and HCF for memory.

Take example, power by taking into the highest power possible

• Tips, McQ = Take example given with ratio or numbers. Important, in this mcqs: you should set examples, try a case for every result given. This involves understanding HCF and LMC definitions*

Real numbers and relationships to factors:

Remember these key examples when coming to solving for a, or c and seeing a relationship.

Given to Remember P and Q are factors, then we have A and p, natural numbers, then we use all properties to solve further: use formula

Take away to write remember: Examples

• P times / is or multiplication = always to all examples

What to expect, example

If they say, q * 3 = P What then is p? A: Factor If you understand examples shown in class, and given above, they you remember it all..

Remember! Follow formula: then plug

Formula : **HCF/LCM by two numbers

Example:

• In mind , follow is to the same concept with other one on doing : example by following it all

Formula for success.

What is another Key equation

How To : Find a solution for factors/LCM (Just ask)

Answer : " Then Follow Method and Ask" == Result

Follow Key questions in class

Key Formula:

• The main key formula to relate

  LMC is multiple : High, + more HCF == Great in factoring, small short.

What needs to be remembered

1 : For every formula == there the 3040 is how you need to remember. 2 : Remember to use formula and write at every step

• Key note/point: Look for main relationship and apply the idea to the formula = makes it eaiser*

Remember that this key value is for two numbers, and also relate the chain: That chain includes HCF times LCM.

What to know about prime factorization.

1: Understand prime numbers

A: Use this formula to find: and understand the step.

Takeaway: Follow what it is a factor of with example

• You could also find an element.*

It has to include the relationship.

What can u derive from what is the core of comp numbers= take

It needs to be "more 2 "

If there a lot

1- Remember key to all questions: Follow main point what do they all have in common. 2- Use that.

More advice to do

If the solution is big and long, show all.

• Keep the side constant by working everything at once
• Chain one into another with formula to make the results better
• When chain it all with previous with next to a relationship = then follow

To solve for irrational

Core Steps to always apply:

• Prove that formula is right
• Key focus: OZI rules
• Use 7 = or 5 to show an example relation

Show everything on what steps are used.

Write short note Show steps to everything that we use

Side steps to help for irrational values

Do the opposite and then prove that it is an opposite, as is core

• If no relation try to chain values between the questions
• Try chain values such how to make = all, equal
• Then use quick steps of the chain to see if it does add up to the same. Important steps:
• Find what is given, with that said always set "what is it needs" Remember
• What is to remember"

Remember

Rational and rrational will result to answer on solving everything

Chain to next step, the use of all of them to the formula

Polynomials: Quick steps to all solve or do

• Define, all, with example
• Do every example and steps with every detail, write formula on side All this you will remember.
• Key value is to use = to zero

Solve quadratic

• Split middle term + other, important:

• Key : Solve split

  = this value, then know Quick

• Always solve to 0, quick to finish if u need

Very important concepts about this

Is what is the root with product in an efficient example.

Take what equation then extract quick.

Is what it means.

Very very important to remember what the process is for all types.

What to do

Quick Note:

Chain to find " that equals", is the KEY For this part you will, follow step by step

All Key

Remember how steps are done always to the end, else remember = do it all over If all that happens Look at all equation used relate. Quick not to remember Then write formula

"Remember to take and plug." You'll have, two.

Tips

Remember and list side note of this in tips Chain What to do Why to do formula" Take what it needs only

That is the process

Very important for McQ

Quick

• When stuck ask"

Quick way Solve what is all in quick and easy steps, so solve quickly

Important McQ question

""Find what is asked then do or remember""

Tips : All

Then do some thing with example relation == answer. Again" Quick steps or notes to do == that can be chain. Or remember with short What does the 2nd part ask.

All : Main tip: Chain!

Quick" To quick finish and find the answers solve in reverse" To remember.

You can also try to solve or look for a short quick and chain relationship method to solving it == for every example

• Example in class does have this way.. Take not. A lot to do you'll have, but if done correctly, you remember it all, " "You all will."