Trigonometry: Class 10 Introduction

Introduction to Trigonometry

• Trigonometry is used for measuring heights and distances, including monuments like the Taj Mahal and even mountains.
• It has applications in music, oceans, and seas.
• It essentially measures heights and distances.

Understanding the Term "Trigonometry"

• The word "trigonometry" combines "trigono" (three-sided polygon, i.e., triangle) and "metry" (measurement).
• Specifically, in Class 10th, trigonometry predominantly deals with right-angled triangles.
• Trigonometry involves measuring triangles, including their three angles and three sides.

Trigonometric Ratios Explained

• In a right-angled triangle, the base and perpendicular are determined relative to the angle in consideration.
• The hypotenuse is always opposite the 90-degree angle.
• The side on which the angle rests is the base.
• The side opposite the angle is the perpendicular.
• Another name for base is adjacent side.
• Another name for the perpendicular is opposite side.
• Trigonometric ratios compare the sides of a right-angled triangle to each other. There are six ratios: sine, cosine, tangent, cosecant, secant, and cotangent.
• These ratios involve comparisons (by division) between two sides of a triangle, providing specific names for each comparison, e.g., sine compares perpendicular to hypotenuse.
• The ratios are:
  • Sine (sin θ) = Perpendicular / Hypotenuse
  • Cosine (cos θ) = Base / Hypotenuse
  • Tangent (tan θ) = Perpendicular / Base
  • Cosecant (cosec θ) = Hypotenuse / Perpendicular (reciprocal of sine)
  • Secant (sec θ) = Hypotenuse / Base (reciprocal of cosine)
  • Cotangent (cot θ) = Base / Perpendicular (reciprocal of tangent)

Important Points About Trigonometric Ratios

• 'Sin θ' is read as "sine of angle theta".
• 'Sin θ' does not mean sin multiplied by theta; it's an indivisible term.
• If 'sin' is written without an angle (e.g., just 'sin'), it’s meaningless in trigonometry.
• Each trigonometric ratio results in a real number but has no units as it's a ratio.

Calculating Trigonometric Ratios

• Given one trigonometric ratio, you can find all others using the Pythagorean theorem. Use
  • h² = p² + b²

True or False Examples

• False: sin θ cannot be 3/2 as hypotenuse > perpendicular. Max value of sin is always 1
• Learning: sin θ is always between 0 and 1 because hypotenuse > perpendicular, therefore it's a fraction (by division)
• True: sec a can be 12/5.
• False: cos a is NOT abbreviation for cosecant

Trigonometric Ratios of Specific Angles

• Focus is on 0°, 30°, 45°, 60°, and 90°.
• You only need to remember the sine values for these angles, rest can be derived.
• Values to Remember:
  • sin 0° = 0
  • sin 30° = 1/2
  • sin 45° = 1/√2
  • sin 60° = √3/2
  • sin 90° = 1
• Cosine values are just sine values in reverse.
• Tangent values are derived from dividing sine/cosine. Undefined means infinity, but Not Defined is most correct term

Solving Problems Involving Specific Angles

• If sin θ = √3/2, then θ = 60°, because sin 60° = √3/2.
• Ensure that while solving these expressions, put correct angle as defined in the question
• On comparison, sin (angle) allows you to understand angle on left side as a function of the angle on right

Trigonometric Identities

• These can be remembered as "one and square वाले formulas"
• Key Identities:
  • sin² θ + cos² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = cosec² θ

Manipulation of Equations

• sin² θ + cos² θ =1:*
• Rearranging this can allow you to express cos in terms of sin (and vice versa): cos² θ = 1 - sin² θ
• 1 + tan² θ = sec² θ:*
• tan² θ = sec2 θ - 1.
• 1 + cot² θ = cosec² θ:*
• cot² θ = cosec² θ - 1.

Examples of Problem-Solving Using Identities

• If asked to convert cos, tan, and sec into sin. Then:
• cos = √(1 - sin²A)
• tan = sin / √(1 - sin²A)
• sec = 1/ √(1 - sin²A)
• You need to be able to manipulate and express various trigonometric functions in terms of other trigonometric functions.

Proof Strategies

• L.H.S = R.H.S with 3 Techniques: 1) L.H.S = R.H.S, 2) R.H.S = L.H.S, 3) Simplify both until equal
• You should start with Left hand side by default!
• Multiply and distribute to yield result, know identities!
• If in doubt, convert each function to sin + cos

Key Guidelines

• Use above identities to guide, and keep options of all options of formulas
• See the trigonometric function of the Right hand side, and try to make changes to trigonometric functions of the Left hand side

Special Cases of Trigonometry

• For 0 < theta <90 - always take positive values
• In particular style of question show, you are given two values, which are connected, the third may be one. Use trick that the teacher discusses:
• You are trying to yield a factor on Left Hand Side = factor on Right Hand Side. So set them both equal, and cancel same terms on both sides.
• Or cancel similar factors via other mathematical operations.

Trigonometric Identities List

• Reciprocal Identity: Need to remember that sin <> cosec travel together, cos <> sec travels together, tan <> cot travel together! Use cases of these are also provided
• Quotient Identity: Remember tan = sin/cos and cot = cos/ sin is important

Helpful Trigonometry Formulas

• ( a^3 + b^3 = (a + b)(a^2 - ab + b^2) )
• ( a^3 - b^3 = (a - b)(a^2 + ab + b^2) )
• If you look at ( (a - b), then use (+ab) ) with the other side, and if you start look at ( (a + b) use (-ab) ) with other side

Trigonometry Techniques and Tricks

• When an equation involves something equal or close to the Left and Right hand side , do what happens on Left hand side, with Right hand side!
• If power 4 is causing issues. See how it can convert to power 2
• If there seems to be something near equation cancel, it means it will cancel. Make a step or two to cancel by using appropriate operations like factorization... or distribution.

Techniques Specific to a Type of Trigonometry Question

• If Left and Right hand side involve trigonomic equation with denominator: Start with Left hand side, rationalize to see if get an immediate equation, cancel top and bottom etc
• Keep above trick and formula in mind to think of ways to start.

If Trigonometry Not Easily Visible and Seems Confusing

• Follow the below general algorithm
• Express all trig function in cos and sin only

Techniques When Near a Solved State For Trigonometric Operations

• Seperate them as applicable like, denominator separation

Memorization Trick for Trigonometry - What Happens to the Value As Theta Increases From 0 to 90?

• Make cos a function that is decreasing
• What is allowed and not allowed
• division by 0 not allowed
• tan increases
• cot decreases

High-level trick for a very particular type of question... Which needs to be memorized

• For a particular question type do these exact steps:*
• Set all formulas to sine and cosine
• Cancel sine and cos and factor them out
• You might then have to use these factorization formula
• Be sure every step makes sense

For Harder Trigonometry Questions

• These can be hard without tricks and require practice, the teacher has used a lot of trigonometry over the years!
• There are certain questions that are only asked now and them, usually when the paper is particularly hard
• When given a 3 function solve and equate that can be factored further
• When near an impossible problem- convert sec, sine, cosine to tan for easier solutions
• For special problem- if you look at the answer over right, convert what is left, to the trigonometric function of the right side. To simplify to that result.

Common Technique for Complex Proof Problems

Add 1, add 1! It works and then you can proceed and see common patterns

Most Important Hard Problems:

Those with power 4 and the equation where the factorization formula are used.

Some More Tricks and Techniques: To Factor

1. You can have 3 sections of problems that have 6 trigonometric functions, then change them to cos and sine to get to answer.

2. To Yield Answer On Right hand side

Type-Specific Notes and Algorithms For SpecialTrigonometry Operations

1. • You have to find solution when 3 trig functions in picture If there are 3 trigonometric functions that are required, then switch to tan.

Very Special Trigonometry Functions Worth Remembering:

You do you really quickly on hard questions: These only sometimes show up like the hardest question... So up to you

• Reminder that:* Sec + \tan \theta =p Sec- \tan \theta = 1 / p Also, cosec and cot.

Algorithm

Remember These Results

Important Memory Reminder

• In trigonometry use sin cosec together always!