Trigonometric Ratios of Complementary Angles
Two angles are complementary if their sum is 90 degrees. The trigonometric ratios of complementary angles have a special relationship: the sine of an angle equals the cosine of its complement, and vice versa.
These relationships are covered in Chapter 8 (Introduction to Trigonometry) of the NCERT Class 10 textbook. They are also called co-function identities because they relate "co-" functions (sin-cos, tan-cot, sec-cosec).
These identities are essential for simplifying trigonometric expressions and evaluating compound expressions involving different angles that sum to 90 degrees.
The word "cosine" literally means "complement's sine" — the cosine of an angle is the sine of its complement. This etymological connection runs through all three reciprocal pairs: co-sine, co-tangent, and co-secant each refer to the corresponding ratio of the complementary angle.
These co-function identities are particularly useful in CBSE board examinations for evaluating expressions without using trigonometric tables. Problems typically present combinations of trigonometric ratios at different angles that sum to 90 degrees, and the student must recognise the complementary relationship to simplify the expression.
The proof of these identities is straightforward and relies on the geometry of a right triangle: the side opposite one acute angle is the side adjacent to the other acute angle, while the hypotenuse remains the same. This geometric insight makes the identities intuitive rather than merely formulaic.
What is Trig Ratios of Complementary Angles - Identities, Proof & Examples?
Complementary Angles: Two angles A and B are complementary if A + B = 90 degrees.
If A + B = 90 degrees, then B = 90 degrees - A.
Co-function Identities:
sin(90 - A) = cos A
cos(90 - A) = sin A
tan(90 - A) = cot A
cot(90 - A) = tan A
sec(90 - A) = cosec A
cosec(90 - A) = sec A
Key insight: The prefix "co-" in cosine, cotangent, and cosecant stands for "complement." The cosine of an angle IS the sine of its complement.
Why are they called "co-function" identities?
- Co-sine = sine of the complement
- Co-tangent = tangent of the complement
- Co-secant = secant of the complement
This naming convention has been part of mathematical terminology for centuries and directly encodes the complementary angle relationships.
Trigonometric Ratios of Complementary Angles Formula
Complementary Angle Identities:
| Identity | Example (A = 30 degrees) |
|---|---|
| sin(90 - A) = cos A | sin 60 = cos 30 = sqrt(3)/2 |
| cos(90 - A) = sin A | cos 60 = sin 30 = 1/2 |
| tan(90 - A) = cot A | tan 60 = cot 30 = sqrt(3) |
| cot(90 - A) = tan A | cot 60 = tan 30 = 1/sqrt(3) |
| sec(90 - A) = cosec A | sec 60 = cosec 30 = 2 |
| cosec(90 - A) = sec A | cosec 60 = sec 30 = 2/sqrt(3) |
Quick rule: When the angle changes from A to (90 - A):
- sin ↔ cos
- tan ↔ cot
- sec ↔ cosec
Derivation and Proof
Proof of Complementary Angle Identities:
Consider a right triangle ABC with angle B = 90 degrees.
- Let angle A = A degrees, then angle C = (90 - A) degrees (since A + C = 90).
- sin A = BC/AC (opposite/hypotenuse for angle A)
- cos C = cos(90 - A) = BC/AC (adjacent/hypotenuse for angle C)
- Therefore: sin A = cos(90 - A)
Similarly:
- cos A = AB/AC (adjacent/hypotenuse for angle A)
- sin C = sin(90 - A) = AB/AC (opposite/hypotenuse for angle C)
- Therefore: cos A = sin(90 - A)
The same reasoning applies to tan-cot and sec-cosec pairs:
- tan A = BC/AB = cot(90 - A)
- sec A = AC/AB = cosec(90 - A)
The proof works because the side that is "opposite" to angle A is "adjacent" to angle C (and vice versa), while the hypotenuse remains the same.
Types and Properties
Types of problems using complementary angle identities:
- Evaluate expressions: sin 55 / cos 35 = sin 55 / cos 35 = sin 55 / sin 55 = 1 (since cos 35 = sin 55).
- Simplify compound expressions: Replace sin(90 - A) with cos A to simplify.
- Prove identities: Convert one or both sides using co-function identities.
- Solve equations: If sin A = cos B, then A + B = 90 degrees.
- Find angle values: tan A = cot B implies A + B = 90 degrees, so if A = 2x and B = 3x, solve 5x = 90.
Common complementary angle pairs in problems:
| Angle A | Complement (90-A) |
|---|---|
| 10 | 80 |
| 15 | 75 |
| 18 | 72 |
| 20 | 70 |
| 25 | 65 |
| 27 | 63 |
| 28 | 62 |
| 35 | 55 |
| 37 | 53 |
| 40 | 50 |
Recognising these pairs quickly is crucial for exam speed.
Methods
Method 1: Direct Substitution
Replace sin(90 - A) with cos A (and similar pairs). Then simplify.
Method 2: Recognizing Complementary Pairs
When you see two angles that sum to 90 degrees (like 37 and 53, or 28 and 62), use the identities to convert one to match the other.
Method 3: Solving Equations
If sin A = cos B, then A + B = 90 degrees. Use this to set up an equation and solve.
Tips:
- Look for angles that add up to 90: 20+70, 25+65, 35+55, 40+50, etc.
- sin/cos are a pair, tan/cot are a pair, sec/cosec are a pair.
- sin 0 = cos 90 = 0, and sin 90 = cos 0 = 1 (check using complementary relationship).
Recognising complementary angle pairs in problems:
- Look for two angles that add to 90: (20, 70), (25, 65), (35, 55), (18, 72), (37, 53), (28, 62), etc.
- In expressions like sin^2(A) + cos^2(90-A), recognise that cos(90-A) = sin A, so the expression becomes sin^2 A + sin^2 A = 2 sin^2 A.
- Products like tan A x cot(90-A) = tan A x tan A = tan^2 A, NOT 1.
- But tan A x tan(90-A) = tan A x cot A = 1 — the second angle must be the complement.
Exam strategy:
- When you see sin or cos of an unusual angle (like 37 degrees), look for cos or sin of its complement (53 degrees) elsewhere in the expression.
- When solving equations, if sin A = cos B, immediately write A + B = 90 degrees.
- When evaluating series like sin^2(5) + sin^2(10) + ... + sin^2(85) + sin^2(90), pair terms from both ends using complementary angles.
Solved Examples
Example 1: Evaluating sin/cos of Complementary Angles
Problem: Evaluate: sin 72 degrees / cos 18 degrees.
Solution:
- cos 18 = sin(90 - 18) = sin 72
- So sin 72 / cos 18 = sin 72 / sin 72 = 1
Answer: 1.
Example 2: Evaluating a Compound Expression
Problem: Evaluate: tan 25 degrees x tan 65 degrees.
Solution:
- tan 65 = tan(90 - 25) = cot 25
- tan 25 x cot 25 = tan 25 x (1/tan 25) = 1
Answer: 1.
Example 3: Simplifying a Trigonometric Expression
Problem: Simplify: cos 48 degrees - sin 42 degrees.
Solution:
- sin 42 = sin(90 - 48) = cos 48
- So cos 48 - sin 42 = cos 48 - cos 48 = 0
Answer: 0.
Example 4: Solving for an Unknown Angle
Problem: If tan 2A = cot(A - 18 degrees), where 2A is an acute angle, find A.
Solution:
Given: tan 2A = cot(A - 18)
Using: tan theta = cot(90 - theta)
- tan 2A = cot(90 - 2A)
- So cot(90 - 2A) = cot(A - 18)
- 90 - 2A = A - 18
- 108 = 3A
- A = 36 degrees
Verification: tan 72 = cot 18. Since 72 + 18 = 90, this is correct.
Answer: A = 36 degrees.
Example 5: Evaluating at Standard Angles
Problem: Evaluate: sin^2(20) + sin^2(70) + cos^2(20) + cos^2(70).
Solution:
- sin 70 = cos 20 and cos 70 = sin 20 (complementary angles)
- sin^2(20) + sin^2(70) = sin^2(20) + cos^2(20) = 1
- cos^2(20) + cos^2(70) = cos^2(20) + sin^2(20) = 1
- Total = 1 + 1 = 2
Answer: 2.
Example 6: Product of Complementary Ratios
Problem: Evaluate: sin 10 x sin 20 x sin 30 x sin 40 x sin 50 x sin 60 x sin 70 x sin 80.
Solution:
Pair complementary angles:
- sin 10 x sin 80 = sin 10 x cos 10
- sin 20 x sin 70 = sin 20 x cos 20
- sin 30 = 1/2
- sin 40 x sin 50 = sin 40 x cos 40
- sin 60 = sqrt(3)/2
Using sin A x cos A = sin 2A/2:
- = (sin 20/2) x (sin 40/2) x (1/2) x (sin 80/2) x (sqrt(3)/2)
This is a complex product. For the exam, recognise the pairing technique and use standard values.
Answer: 3/256.
Example 7: Finding sin A + cos A
Problem: If sin(A - B) = 1/2 and cos(A + B) = 1/2, where A > B, find A and B.
Solution:
- sin(A - B) = 1/2 → A - B = 30 degrees
- cos(A + B) = 1/2 → A + B = 60 degrees
Adding: 2A = 90, so A = 45 degrees
Subtracting: 2B = 30, so B = 15 degrees
Answer: A = 45 degrees, B = 15 degrees.
Example 8: Proving an Identity with Complementary Angles
Problem: Prove: sin(90 - A) x cosec(90 - A) = 1.
Solution:
LHS:
- sin(90 - A) = cos A
- cosec(90 - A) = sec A = 1/cos A
- LHS = cos A x (1/cos A) = 1 = RHS
Hence proved.
Example 9: Trigonometric Table Verification
Problem: Without using tables, evaluate: (cos 80)/(sin 10) + cos 59 x cosec 31.
Solution:
- cos 80 = cos(90 - 10) = sin 10
- cos 80/sin 10 = sin 10/sin 10 = 1
- cosec 31 = cosec(90 - 59) = sec 59
- cos 59 x sec 59 = cos 59 x (1/cos 59) = 1
- Total = 1 + 1 = 2
Answer: 2.
Example 10: Complementary Angles in a Right Triangle
Problem: In a right triangle ABC with angle B = 90, show that sin A = cos C and tan A = cot C.
Solution:
Since A + C = 90 degrees (angle sum in a triangle, and B = 90):
- C = 90 - A
- cos C = cos(90 - A) = sin A. Hence sin A = cos C.
- cot C = cot(90 - A) = tan A. Hence tan A = cot C.
Hence proved.
Example 11: Evaluating a Sum of Squares
Problem: Evaluate: sin^2 5 + sin^2 10 + sin^2 15 + ... + sin^2 85 + sin^2 90.
Solution:
Pair complementary terms:
- sin^2 5 + sin^2 85 = sin^2 5 + cos^2 5 = 1
- sin^2 10 + sin^2 80 = sin^2 10 + cos^2 10 = 1
- sin^2 15 + sin^2 75 = 1
- sin^2 20 + sin^2 70 = 1
- sin^2 25 + sin^2 65 = 1
- sin^2 30 + sin^2 60 = 1
- sin^2 35 + sin^2 55 = 1
- sin^2 40 + sin^2 50 = 1
That gives 8 pairs, each summing to 1.
Remaining: sin^2 45 = (1/sqrt(2))^2 = 1/2, and sin^2 90 = 1.
Total = 8 x 1 + 1/2 + 1 = 8 + 1.5 = 9.5 = 19/2.
Answer: 19/2.
Example 12: Solving a Complementary Angle Equation with sec and cosec
Problem: If sec 5A = cosec(A + 36), where 5A is an acute angle, find A.
Solution:
Using: sec theta = cosec(90 - theta)
- sec 5A = cosec(90 - 5A)
- cosec(90 - 5A) = cosec(A + 36)
- 90 - 5A = A + 36
- 54 = 6A
- A = 9 degrees
Verification: sec 45 = sqrt(2) and cosec 45 = sqrt(2). Since 5(9) = 45 and 9 + 36 = 45, both sides equal cosec 45 = sqrt(2). Verified.
Answer: A = 9 degrees.
Example 13: Simplifying a Multi-Term Expression
Problem: Simplify: cot 12 x cot 38 x cot 52 x cot 60 x cot 78.
Solution:
Pair complementary angles:
- cot 12 x cot 78 = cot 12 x cot(90-12) = cot 12 x tan 12 = 1
- cot 38 x cot 52 = cot 38 x cot(90-38) = cot 38 x tan 38 = 1
- Remaining: cot 60 = 1/sqrt(3)
Result = 1 x 1 x 1/sqrt(3) = 1/sqrt(3) = sqrt(3)/3.
Answer: 1/sqrt(3).
Real-World Applications
Applications of complementary angle identities:
- Simplifying expressions: Reduce complex multi-angle expressions by pairing complementary angles.
- Evaluating without tables: Find values like cos 72 by converting to sin 18 (if sin 18 is known).
- Solving equations: Equations like sin A = cos B immediately give A + B = 90.
- Heights and distances: The angle of elevation from one point equals the complement of the angle of depression from another point in certain configurations.
- Physics: Projectile motion problems use complementary angle pairs (angles of launch for same range).
- Navigation: Bearing conversions between compass headings and mathematical angles use complementary relationships.
Key Points to Remember
- Complementary angles sum to 90 degrees.
- sin(90 - A) = cos A and cos(90 - A) = sin A.
- tan(90 - A) = cot A and cot(90 - A) = tan A.
- sec(90 - A) = cosec A and cosec(90 - A) = sec A.
- The "co-" prefix means "complement": co-sine = sine of the complement.
- If sin A = cos B, then A + B = 90 degrees.
- If tan A = cot B, then A + B = 90 degrees.
- sin^2 A + sin^2(90 - A) = sin^2 A + cos^2 A = 1.
- tan A x tan(90 - A) = tan A x cot A = 1.
- These identities are proved using the fact that in a right triangle, the side opposite one acute angle is adjacent to the other.
Practice Problems
- Evaluate: sin 35 / cos 55.
- Without tables, find: cos^2 27 + cos^2 63.
- If sec 4A = cosec(A - 20), find A.
- Evaluate: tan 5 x tan 25 x tan 45 x tan 65 x tan 85.
- Simplify: sin(90 - A) x sec(90 - A) x tan A / (cot(90 - A) x sin A x cosec A).
- If cos(A - B) = sqrt(3)/2 and sin(A + B) = sqrt(3)/2, find A and B.
- Prove: cos A x cos(90 - A) - sin A x sin(90 - A) = 0.
- Evaluate: sin^2 5 + sin^2 10 + sin^2 15 + ... + sin^2 85 + sin^2 90.
Frequently Asked Questions
Q1. What are complementary angles?
Two angles are complementary if they add up to 90 degrees. For example, 30 and 60 degrees are complementary.
Q2. What is the relationship between sin and cos of complementary angles?
sin(90 - A) = cos A, and cos(90 - A) = sin A. The sine of an angle equals the cosine of its complement.
Q3. Why is cosine called 'cosine'?
Cosine means 'complement's sine.' cos A = sin(90 - A), i.e., the cosine of angle A is the sine of A's complement.
Q4. How do you use complementary angles to evaluate expressions?
Identify pairs of angles that sum to 90. Replace one ratio with its co-function. For example, cos 35 = sin 55, so sin 55/cos 35 = sin 55/sin 55 = 1.
Q5. If tan A = cot B, what can you conclude?
Since tan A = cot(90 - A), and tan A = cot B, we get B = 90 - A, i.e., A + B = 90 degrees.
Q6. What is the value of tan 1 x tan 2 x ... x tan 89?
Pair complementary angles: tan k x tan(90-k) = tan k x cot k = 1 for each pair. The middle term is tan 45 = 1. So the product = 1.
Q7. Are complementary angle identities important for CBSE board exams?
Yes. Questions on evaluating expressions using complementary angles, solving equations, and proving identities are common in CBSE Class 10 board exams.
Q8. Can these identities be extended beyond 90 degrees?
In Class 10, angles are restricted to 0-90 degrees. In Class 11/12, the identities are extended using the unit circle for all angles.
Q9. How do complementary angles relate to a right triangle?
In a right triangle, the two acute angles always sum to 90 degrees (since the third angle is 90). So the two acute angles are always complementary. This is why sin of one angle equals cos of the other.
Q10. Can complementary angle identities be used for angles greater than 90?
In Class 10 NCERT, these identities apply to acute angles (0 to 90 degrees) only. In higher classes, the concept extends using the unit circle to handle all angles.
