Class 12 Mathematics TRIGONOMETRIC EQUATIONS AND GENERAL VALUES Notes

Class 12 Mathematics TRIGONOMETRIC EQUATIONS AND GENERAL VALUES Notes
Class 12 Mathematics Trigonometric Equations and General Values Notes | Nepal eNotes

Unit 9

Trigonometry

Class 12 Mathematics

Trigonometric Equations and General Values

Class 12 Mathematics – Trigonometric Equations and General Values Notes PDF

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Contents / Quick Navigation

1. Trigonometric Equations 2. Periodicity and General Values 3. General Solution of Sine Equations 4. General Solution of Cosine Equations 5. General Solution of Tangent Equations 6. Standard General-Value Formulae 7. Equations Reducible to Standard Form 8. Factorization and Identities 9. Solutions in a Given Interval 10. Worked Examples 11. Common Mistakes 12. Important Exam Questions 13. One-Minute Revision 14. Diagram Practice

NEB / CDC Focus

The main learning goal in this chapter is to find the general solution of trigonometric equations. This requires understanding periodicity, converting equations to standard forms, and expressing all possible angle values using an integer parameter.

1. Trigonometric Equations

Definition

An equation containing one or more trigonometric functions of an unknown angle is called a trigonometric equation.

Examples:

  • sin x=1/2
  • 2cos²x−1=0
  • tan 2x=√3
  • sin x+sin 2x=0

Unlike most elementary algebraic equations, trigonometric equations often have infinitely many solutions because sine, cosine and tangent are periodic.

Why Trigonometric Equations Have Repeated Solutions sin x=1/2 The same sine value appears again after each period of 2π.
Figure 1: Periodicity creates infinitely many solutions unless an interval is specified.

2. Periodicity and General Values

Periodic Function

A function f is periodic with period T if f(x+T)=f(x) for every allowed x, and T>0 is a period.

FunctionFundamental periodPeriodicity relation
sin x2πsin(x+2nπ)=sin x
cos x2πcos(x+2nπ)=cos x
tan xπtan(x+nπ)=tan x
cot xπcot(x+nπ)=cot x

Here and throughout the chapter, n∈ℤ means that n may be any integer: …,−2,−1,0,1,2,….

General Value

A general value is a formula containing an integer parameter that represents every angle satisfying the given trigonometric condition.

Periods Used in General Solutions sin x period 2π repeat with 2nπ cos x period 2π repeat with 2nπ tan x period π repeat with nπ The correct period must appear in the final general solution.
Figure 2: Sine/cosine repeat every 2π; tangent repeats every π.

3. General Solution of Sine Equations

3.1 Equation sin x = sin α

Sine has the same value at angles α and π−α in one full cycle. Repeating those values gives:

General Solution
sin x = sin α   ⇒   x=nπ+(−1)nα,   n∈ℤ.

An equivalent two-family form is

x=α+2nπ or   x=π−α+2nπ,   n∈ℤ.
Why sin α = sin(π−α) α π−α same y-coordinate → same sine
Figure 3: Angles α and π−α have equal sine values.
Worked Example

Solve sin x=1/2 generally.

Since sin(π/6)=1/2,

x=nπ+(−1)nπ/6,   n∈ℤ.

Equivalent:

x=π/6+2nπ   or   x=5π/6+2nπ.

4. General Solution of Cosine Equations

4.1 Equation cos x = cos α

Cosine has the same value at α and −α. Therefore:

General Solution
cos x=cos α   ⇒   x=2nπ±α,   n∈ℤ.
Why cos α = cos(−α) α −α same x-coordinate → same cosine
Figure 4: Angles α and −α have equal cosine values.
Worked Example

Solve cos x=√3/2 generally.

Since cos(π/6)=√3/2,

x=2nπ±π/6,   n∈ℤ.

5. General Solution of Tangent Equations

5.1 Equation tan x = tan α

Tangent repeats after π. Hence:

General Solution
tan x=tan α   ⇒   x=nπ+α,   n∈ℤ.

Similarly,

cot x=cot α   ⇒   x=nπ+α,   n∈ℤ.
Tangent Repeats Every π −π/2 π/2 Moving an angle by π leaves its tangent unchanged.
Figure 5: The fundamental period of tan x is π.
Worked Example

Solve tan x=1 generally.

Since tan(π/4)=1,

x=nπ+π/4,   n∈ℤ.

6. Standard General-Value Formulae

EquationGeneral solution
sin x=sin αx=nπ+(−1)ⁿα
cos x=cos αx=2nπ±α
tan x=tan αx=nπ+α
cot x=cot αx=nπ+α
sin x=0x=nπ
cos x=0x=(2n+1)π/2
tan x=0x=nπ
sin x=1x=(4n+1)π/2
sin x=−1x=(4n−1)π/2
cos x=1x=2nπ
cos x=−1x=(2n+1)π
Exam Important

Always write n∈ℤ with a general solution. Without the integer parameter, the answer normally gives only selected values rather than all solutions.

7. Equations Reducible to Standard Form

7.1 Multiple-Angle Equations

If sin(mx)=sin α, first solve for the angle mx, then divide the entire general solution by m.

Worked Example: sin 2x=sin π/3

Using the sine general form:

2x=nπ+(−1)ⁿπ/3.

Therefore

x=nπ/2+(−1)ⁿπ/6,   n∈ℤ.

7.2 Equations of the Form a sin x + b cos x = 0

If division is valid,

a sin x+b cos x=0   ⇒   tan x=−b/a

when a≠0 and no solutions are lost by the chosen division.

Example

Solve √3 sin x−cos x=0.

√3 sin x=cos x ⇒ tan x=1/√3.

x=nπ+π/6,   n∈ℤ.

7.3 Quadratic Trigonometric Form

Treat equations such as 2sin²x−3sin x+1=0 as quadratic equations in sin x.

Example

Solve 2sin²x−3sin x+1=0.

(2sin x−1)(sin x−1)=0.

Thus

sin x=1/2   or   sin x=1.

Hence combine the general values of both equations.

Reduce → Solve Standard Form → Generalize Original equation identities / algebra simplify first Standard equation sin u=sinα cos u=cosα / tan u=tanα General solution n∈ℤ Do not memorize a separate formula for every complicated equation; reduce it first.
Figure 6: Most exam problems become standard after suitable algebra or identities.

8. Solving by Factorization and Trigonometric Identities

8.1 Factorization

Worked Example

Solve sin x cos x=0.

A product is zero if at least one factor is zero:

sin x=0   or   cos x=0.

x=nπ   or   x=(2n+1)π/2.

These two families together are equivalently x=nπ/2, n∈ℤ.

8.2 Using Double-Angle Identities

Worked Example

Solve 2sin x cos x=1.

Since 2sin x cos x=sin2x,

sin2x=1.

2x=(4n+1)π/2.

x=(4n+1)π/4,   n∈ℤ.

8.3 Converting cos 2x

Useful Identities
sin2x=2sin x cos x cos2x=cos²x−sin²x=1−2sin²x=2cos²x−1 1−cos2x=2sin²x 1+cos2x=2cos²x

8.4 Sum-to-Product / Product-to-Sum When Useful

Some equations simplify by transforming a sum into a product and then setting factors equal to zero.

sin A+sin B=2sin((A+B)/2)cos((A−B)/2) sin A−sin B=2cos((A+B)/2)sin((A−B)/2) cos A+cos B=2cos((A+B)/2)cos((A−B)/2) cos A−cos B=−2sin((A+B)/2)sin((A−B)/2)
Example

Solve sin3x+sin x=0.

2sin2x cos x=0.

Therefore

sin2x=0   or   cos x=0.

The first gives x=nπ/2; the second is already included in that family. Thus

x=nπ/2, n∈ℤ.

9. Solutions in a Given Interval

A general solution represents all possible solutions. If the question instead asks for values in an interval such as 0≤x<2π, first obtain a correct general family and then choose only the values lying inside the interval.

Worked Example: Solve sin x=−1/2 for 0≤x<2π

Reference angle is π/6. Sine is negative in quadrants III and IV.

x=7π/6,   11π/6.

Quadrant Signs Help with Interval Solutions I sin + cos + tan + II sin + III tan + IV cos + Use the sign together with the reference angle to list interval-specific solutions.
Figure 7: Quadrant signs are useful when converting a general relation into solutions on one cycle.
Endpoint Check

Distinguish carefully between intervals such as 0≤x<2π, 0<x≤2π, and 0≤x≤2π. The endpoints may change the answer set.

10. Worked Examples

Example 1: cos 2x=cos π/3

Using cos u=cosα ⇒ u=2nπ±α,

2x=2nπ±π/3.

x=nπ±π/6,   n∈ℤ.

Example 2: tan 3x=tan π/4

3x=nπ+π/4.

x=nπ/3+π/12,   n∈ℤ.

Example 3: 2cos²x−1=0

2cos²x−1=cos2x.

Thus

cos2x=0.

2x=(2n+1)π/2 ⇒ x=(2n+1)π/4.

Example 4: sin²x=3/4

sin x=±√3/2.

Equivalently, since cos2x=1−2sin²x,

cos2x=1−3/2=−1/2.

Either route gives the full solution family after applying the standard general-value formula.

Example 5: 2sin x=1

sin x=1/2.

x=nπ+(−1)ⁿπ/6,   n∈ℤ.

Example 6: 2cos x+1=0

cos x=−1/2=cos(2π/3).

x=2nπ±2π/3,   n∈ℤ.

Example 7: sin x=cos x

If cos x≠0, divide by cos x:

tan x=1.

x=nπ+π/4,   n∈ℤ.

Values with cos x=0 do not satisfy the original equation, so no solutions were lost.

Example 8: 2sin²x−sin x−1=0

Factor:

(2sin x+1)(sin x−1)=0.

Hence

sin x=−1/2   or   sin x=1.

Write the general values for both cases and combine them.

Example 9: Solve in 0≤x<2π

Solve 2cos x=−√2.

cos x=−√2/2.

Reference angle is π/4. Cosine is negative in quadrants II and III:

x=3π/4,   5π/4.

Example 10: sin2x=0 in 0≤x≤2π

2x=nπ ⇒ x=nπ/2.

For 0≤x≤2π, choose n=0,1,2,3,4.

x=0, π/2, π, 3π/2, 2π.

11. Step-by-Step Strategy

  1. Simplify first: use algebra or identities to reduce the equation.
  2. Factor if possible: a product equal to zero creates simpler equations.
  3. Identify a standard form: sine, cosine, tangent or cotangent.
  4. Use the correct general-value formula.
  5. For multiple angles: solve for the full angle first, then divide.
  6. Write n∈ℤ for general solutions.
  7. For a specified interval: select only values lying within the endpoints.
  8. Check forbidden divisions: if dividing by sin x or cos x, verify that solutions where the divisor is zero are not lost.
  9. Substitute representative values when unsure about a sign or branch.

12. Common Mistakes and Warnings

Mistake 1: Writing only principal values

A general-solution question needs every periodic solution, not just one angle.

Mistake 2: Wrong period for tan

Tangent repeats after π, not 2π.

Mistake 3: Forgetting the second sine family

sin x=sinα generally has two solutions in a 2π cycle unless they coincide.

Mistake 4: Wrong cosine sign family

Use x=2nπ±α for cos x=cosα.

Mistake 5: Dividing only one part of a multiple-angle result

After solving for 2x or 3x, divide the entire general expression.

Mistake 6: Losing solutions by division

Before dividing by sin x or cos x, check whether the divisor can be zero.

Mistake 7: Ignoring interval endpoints

Check whether each endpoint is included or excluded.

Mistake 8: Degrees and radians mixed

Use one angle unit consistently throughout a solution.

13. Exam-Important Formula Sheet

Core General Values
sin x=sinα ⇒ x=nπ+(−1)ⁿα Equivalent: x=α+2nπ or x=π−α+2nπ cos x=cosα ⇒ x=2nπ±α tan x=tanα ⇒ x=nπ+α cot x=cotα ⇒ x=nπ+α sin x=0 ⇒ x=nπ cos x=0 ⇒ x=(2n+1)π/2 tan x=0 ⇒ x=nπ n∈ℤ in every general solution

14. Important Exam Questions

Short-Answer Questions

  1. Define a trigonometric equation.
  2. What is meant by the general value or general solution of a trigonometric equation?
  3. State the general solution of sin x=sinα.
  4. State the general solution of cos x=cosα.
  5. State the general solution of tan x=tanα.
  6. Write the general values for sin x=0 and cos x=0.
  7. State the fundamental periods of sine, cosine and tangent.

Derivation / Explanation Questions

  1. Explain why sin x=sinα gives the two families x=α+2nπ and x=π−α+2nπ.
  2. Explain geometrically why cos x=cosα gives x=2nπ±α.
  3. Explain why the general solution of tan x=tanα uses nπ.

Numerical / Application Questions

  1. Find the general solution of a basic sine equation.
  2. Find the general solution of a basic cosine equation.
  3. Find the general solution of a tangent equation.
  4. Solve a multiple-angle equation such as sin2x=c or cos3x=c.
  5. Solve a quadratic equation in sin x or cos x.
  6. Solve an equation by factorization after using a trigonometric identity.
  7. Find all solutions in a specified interval such as 0≤x<2π.
  8. Solve equations involving sums such as sinA+sinB=0 after transforming them to product form.

Diagram / Graph Questions

  1. Use a sine graph to show why the same sine value repeats.
  2. Use a unit circle to show that sinα=sin(π−α).
  3. Use a unit circle to show that cosα=cos(−α).
  4. Sketch tangent over successive periods to show that its period is π.
  5. Use a quadrant-sign diagram when solving over one full cycle.

15. One-Minute Revision

Quick Revision
  • Trigonometric equations usually have infinitely many solutions because trig functions are periodic.
  • Sine and cosine have fundamental period 2π.
  • Tangent and cotangent have fundamental period π.
  • sin x=sinα ⇒ x=nπ+(−1)ⁿα.
  • Equivalent sine form: x=α+2nπ or x=π−α+2nπ.
  • cos x=cosα ⇒ x=2nπ±α.
  • tan x=tanα ⇒ x=nπ+α.
  • sin x=0 ⇒ x=nπ.
  • cos x=0 ⇒ x=(2n+1)π/2.
  • Always write n∈ℤ with a general solution.
  • For multiple angles, solve for the entire angle first and then divide.
  • Quadratic trig equations can often be factored like ordinary algebraic quadratics.
  • Use identities to reduce complicated equations to standard forms.
  • When an interval is given, choose only the permitted members of the general families.
  • Never divide by a trig expression without checking whether zero values may be valid solutions.

16. Diagram Practice

  • Sine graph showing repeated intersections with a horizontal line.
  • Period comparison diagram for sine, cosine and tangent.
  • Unit-circle proof that sinα=sin(π−α).
  • Unit-circle proof that cosα=cos(−α).
  • Tangent graph demonstrating period π.
  • Reduction flowchart from a complicated equation to standard general form.
  • Quadrant sign diagram for interval solutions.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
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