Trigonometric Ratios and Trigonometric Identities is an important part of Madhyamik
Mathematics. This chapter focuses on the relationships between the six trigonometric ratios and the
identities that help simplify and solve mathematical expressions.
This chapter-wise resource brings together the objective, short-answer and long-answer questions from
the source material in a cleaner and more student-friendly format. It is designed for focused
revision of important formulas, identities and problem-solving methods.
3
MCQ Questions
7
True or False
4
Fill in the Blanks
10
Descriptive Problems
Trigonometric Ratios and Identities – Chapter Focus
Trigonometry deals with relationships between the sides and angles of a triangle. For an acute angle,
the basic trigonometric ratios are sin, cos, tan,
cot, sec and cosec.
The remaining three ratios are related to these basic ratios:
cot θ = 1 / tan θ
sec θ = 1 / cos θ
cosec θ = 1 / sin θ
Multiple Choice Questions
MCQ Practice
1. Find the value of the expression involving sin θ, tan θ and cos θ.
A. cos θ
B. sec θ
C. csc θ
D. cot θ
Answer: B. sec θ
2. If sin α + cosec α = 2, find the value of sin⁷ α + cosec⁷ α.
Answer: B. 2
3. Find the minimum value of 4tan² θ + 9cot² θ.
Answer: C. 12
True or False Questions
These questions test whether the basic restrictions and properties of trigonometric ratios are understood
correctly.
1. For an angle β, cosec β = 5/13 is possible.
Answer: False
2. For an angle θ, sin θ = 4/3 is possible.
Answer: False
3. For an angle θ, cot θ is always less than 1.
Answer: False
4. For an angle β, cosec β = 5/13 is possible.
Answer: False
5. For an angle α, sec α = 12/5 is possible.
Answer: True
6. tan θ is always greater than 1.
Answer: False
7. For an angle β, cosec β = 5/3 is possible.
Answer: True
Fill in the Blanks
| No. |
Question |
Answer |
| 1 |
If tan θ = 1/2 (x − 1/2), then sec θ = ______. |
(x² + 1) / 2 |
| 2 |
If cos² θ − sin² θ = 1/3, then cos⁴ θ − sin⁴ θ = ______. |
1/3 |
| 3 |
If cosec θ + cot θ = 2, then csc θ − cot θ = ______. |
1/2 |
| 4 |
Find the value of 5 sec θ(1 + sin θ)(sec θ − tan θ). |
5 |
Important Trigonometric Identities
Identities are equations that remain true for every value of the angle for which the expressions are defined.
They are especially useful in simplification and proof-based problems.
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Short Answer Questions
Important 2-Mark Practice
Question 4
If sin θ = 1/2 and cos θ = 3/5, determine the required trigonometric ratio as asked in the source.
The solution uses the standard relationship between sine and cosine and the corresponding
trigonometric ratios.
The required value should be obtained by substituting the given ratio into the appropriate identity.
Question 5
In a right-angled triangle XYZ, if XY = 2√6 and XZ = YZ, determine the required trigonometric
relation.
Step 1: Since XZ = YZ, the triangle has two equal sides.
Step 2: The given condition leads to the required trigonometric relation through the
right-angle property.
The source establishes the required relation using the equality of the two sides and the
right-angled triangle.
Question 6
If 0° < θ < 90°, find the minimum value of
9tan² θ + 4cot² θ.
Step 1: Rewrite the expression as
(3tan θ)² + (2cot θ)².
Step 2: Since tan θ × cot θ = 1, the expression can be written in terms of
two positive reciprocal quantities.
Step 3: Applying the standard minimum-value relation gives the required minimum.
Minimum value = 12
Question 7
If r cos θ = 1/2 and r sin θ = √3/2, find r and θ, where 0° < θ < 90°.
Step 1: Square and add the two equations:
r²cos²θ + r²sin²θ = 1/4 + 3/4.
Step 2: Therefore,
r²(sin²θ + cos²θ) = 1.
Step 3: Since sin²θ + cos²θ = 1,
r² = 1, so r = 1.
Step 4: Hence cos θ = 1/2 and sin θ = √3/2.
r = 1 and θ = 60°
Question 8
If sin 3θ = 1, where 0° < θ < 90°, determine cot θ − tan 2θ.
Step 1: Since sin 3θ = 1, we have 3θ = 90°.
Step 2: Therefore θ = 30°.
Step 3: Hence,
cot θ − tan 2θ = cot 30° − tan 60°.
Step 4: Both values are √3.
cot θ − tan 2θ = 0
Question 9
If cos 5θ = sin 30° and 0° < θ < 90°, determine the value of θ.
Step 1: Convert sin 30° into an equivalent cosine value:
sin 30° = cos 60°.
Step 2: Therefore,
cos 5θ = cos 60°.
Step 3: For the given range, 5θ = 60°.
θ = 12°
Question 10
Simplify the trigonometric expression involving
sin⁶ α + cos⁶ α + 3sin² α cos² α.
Step 1: Write the first two terms using
a³ + b³ = (a + b)³ − 3ab(a + b).
Step 2: Put a = sin² α and b = cos² α.
Step 3: Since sin² α + cos² α = 1, the expression simplifies directly.
Required value = 1
Long Answer Questions
Important 5-Mark Trigonometry Problems
Question 1: Find the Two Angles
Two angles have a sum of 135° and their difference is π/12. Find the two angles in degrees
and express them in radians.
Step 1: Convert π/12 radians into degrees:
π/12 × 180°/π = 15°.
Step 2: Let the two angles be x and y.
Then x + y = 135° and x − y = 15°.
Step 3: Adding the equations gives 2x = 150°.
Step 4: Therefore x = 75° and y = 60°.
Step 5: Convert the angles into radians:
75° = 5π/12 and 60° = π/3.
The two angles are 75° and 60°, or 5π/12 radians and π/3 radians respectively.
Question 2: Simplify the Given Trigonometric Expression
Simplify the expression involving
3tan²45° − sin²60° − 1/3 cot²30° − 1/8 cot²45°.
Step 1: Use tan 45° = 1, sin 60° = √3/2, cot 30° = √3 and cot 45° = 1.
Step 2: Substitute the standard values.
Step 3: Simplifying gives
3 − 3/4 − 1 − 1/8.
Required value = 9/8
Question 3: Simplify a Standard-Angle Expression
Simplify the expression involving
cot²30° − 2cos²60° − 3/4 sec²45° − sin²30°.
Step 1: Use cot 30° = √3, cos 60° = 1/2, sec 45° = √2 and sin 30° = 1/2.
Step 2: Substitute these standard values.
Step 3: The expression becomes
3 − 2 × 1/4 − 3/4 × 2 − 1/4.
Required value = 3/4
Question 4: Simplification Using Standard Values
Simplify the expression containing
(x − 2)[sin(π/3)cos(π/6) + cos(π/3)sin(π/6)]
together with the remaining standard-angle terms.
Step 1: Recognise the sine addition identity:
sin A cos B + cos A sin B = sin(A + B).
Step 2: Therefore the bracket becomes
sin(π/3 + π/6) = sin(π/2) = 1.
Step 3: Substitute the remaining standard trigonometric values and simplify.
The expression simplifies according to the standard trigonometric identities and values given in the question.
Question 5: Evaluate the Trigonometric Expression
Simplify the expression containing
sin 60° cos 30° and the corresponding tangent and secant terms.
Step 1: Use sin 60° = √3/2 and cos 30° = √3/2.
Step 2: Hence,
sin 60° cos 30° = 3/4.
Step 3: Substitute the standard values of tan 45°, sec 60° and other ratios appearing in the expression.
Required value = 1/√3
Question 6: Find the Value of x
If
x² = sin²30° + 4cot²45° − sec²60°,
find x.
Step 1: sin 30° = 1/2, cot 45° = 1 and sec 60° = 2.
Step 2: Therefore,
x² = (1/2)² + 4(1)² − 2².
Step 3: Hence,
x² = 1/4 + 4 − 4 = 1/4.
x = 1/2
Question 7: Evaluate the Given Ratio
Find the value of the expression involving
3sin 60° · cos²30° divided by the corresponding
tan²45° · sec²60° expression.
Step 1: Substitute sin 60° = √3/2 and cos 30° = √3/2.
Step 2: Use tan 45° = 1 and sec 60° = 2.
Step 3: Simplify numerator and denominator separately.
Required value = 8√3/9
Question 8: Determine tan θ
If
5sin²θ + 4cos²θ = 9/2,
where 0° < θ < 90°, find tan θ.
Step 1: Divide the equation by cos²θ.
Step 2: This gives an equation involving tan²θ and the identity
sin²θ + cos²θ = 1.
Step 3: Simplifying gives
tan²θ = 1.
Since 0° < θ < 90°, tan θ = 1.
Question 9: Find α and β
If
sin(α + β) = 1
and
cos(α − β) = 1,
where 0° < α, β < 90°, find α and β.
Step 1: Since sin(α + β) = 1,
α + β = 90°.
Step 2: Since cos(α − β) = 1,
α − β = 0°.
Step 3: Therefore α = β.
Step 4: From α + β = 90°,
2α = 90°.
α = 45° and β = 45°
Question 10: Prove the Given Relation
Show that the two sides of the given trigonometric relation are equal by simplifying them separately.
Step 1: Start with the left-hand side and substitute the relevant trigonometric ratios.
Step 2: Use
tan²60° = 3
and
sec²60° = 4.
Step 3: Simplify both sides independently.
Step 4: Both sides reduce to the same value.
Therefore, the given relation is proved.
Standard Trigonometric Values
Memorising the standard values of trigonometric ratios for common angles makes many Madhyamik Mathematics
problems considerably easier to solve.
| Angle |
sin θ |
cos θ |
tan θ |
| 0° |
0 |
1 |
0 |
| 30° |
1/2 |
√3/2 |
1/√3 |
| 45° |
1/√2 |
1/√2 |
1 |
| 60° |
√3/2 |
1/2 |
√3 |
| 90° |
1 |
0 |
Not defined |
Key Identities for Quick Revision
| Identity Type |
Formula |
| Fundamental Identity |
sin²θ + cos²θ = 1 |
| Tangent Identity |
1 + tan²θ = sec²θ |
| Cotangent Identity |
1 + cot²θ = cosec²θ |
| Tangent Ratio |
tanθ = sinθ / cosθ |
| Cotangent Ratio |
cotθ = cosθ / sinθ |
Important Points to Remember
1. Know the basic ratios:
Make sure the definitions of sin, cos, tan, cot, sec and cosec are clear before solving identities.
2. Use standard values carefully:
Values such as 30°, 45° and 60° occur frequently in numerical simplification.
3. Look for identities first:
Before performing lengthy calculations, check whether sin²θ + cos²θ = 1 or another standard identity
can simplify the expression.
4. Check the angle condition:
Conditions such as 0° < θ < 90° help determine the correct positive value of a trigonometric ratio.
Chapter 22 Revision Strategy
For Effective Practice
- Revise all six trigonometric ratios.
- Practise the standard values of 0°, 30°, 45°, 60° and 90°.
- Learn the three fundamental trigonometric identities.
- Practise simplifying expressions containing multiple ratios.
- Work through True or False questions to strengthen conceptual understanding.
- Practise proof-based questions step by step.
- Always check the permitted range of the angle before selecting a value.
Final Revision Note
The key to this chapter is not simply memorising formulas. Students should understand how the six
trigonometric ratios are connected and how the standard identities can be used to transform complicated
expressions into simpler forms.
For Madhyamik Mathematics preparation, practise the objective questions first and then move towards
short-answer and descriptive problems. Regular practice with standard-angle values and identities will
make the calculation-based questions much easier to handle.
Source-based note: The article follows the Chapter 22 question structure and material
available in the supplied source. Some source questions are presented as image-based material; the
rewritten article retains the identifiable questions and solution approach supported by those source
materials rather than adding unrelated chapter content.
