Trigonometric Functions Solutions

Understanding Trigonometric Functions and Their Applications

Trigonometric functions form the cornerstone of advanced mathematics, bridging the gap between geometry and calculus. These fundamental mathematical tools, derived from right-angled triangles, extend far beyond basic angle measurements to encompass complex mathematical modeling, physics applications, and engineering solutions. Whether you're studying for competitive exams or building a strong foundation in mathematics, mastering trigonometric functions solutions is essential for academic excellence.

The Foundation: Six Fundamental Trigonometric Ratios

In the realm of trigonometry, every concept stems from the relationship between angles and sides in a right-angled triangle. Consider a right triangle ABC where angle C equals 90 degrees. The six primary trigonometric ratios emerge from this simple geometric configuration, each serving a unique purpose in mathematical problem-solving. The sine function represents the ratio of the opposite side to the hypotenuse, while cosine expresses the adjacent side to hypotenuse ratio. The tangent function, fundamentally linked to both sine and cosine, equals their quotient. These three primary functions give rise to their reciprocals: cosecant, secant, and cotangent, creating a complete system for angle analysis.

Essential Trigonometric Identities and Their Power

The beauty of trigonometric functions lies in their interconnected relationships, expressed through fundamental identities that serve as powerful problem-solving tools. The Pythagorean identity, sin²A + cos²A = 1, stands as the most fundamental relationship, from which numerous other identities derive. This primary identity transforms into related expressions: 1 + tan²A = sec²A and cot²A + 1 = cosec²A, each offering unique advantages in different mathematical contexts. These identities aren't merely academic exercises; they represent the underlying harmony in trigonometric relationships, enabling elegant solutions to complex problems that would otherwise require extensive calculations.

Important Trigonometric Formulas Reference Table

Formula Category	Mathematical Expression	Explanation
Pythagorean Identity	sin²A + cos²A = 1	Fundamental relationship between sine and cosine
Tangent Identity	1 + tan²A = sec²A	Relates tangent to secant function
Cotangent Identity	cot²A + 1 = cosec²A	Links cotangent with cosecant
Double Angle (Sine)	sin 2A = 2 sin A cos A	Sine of double angle formula
Double Angle (Cosine)	cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A	Multiple forms for cosine of double angle
Double Angle (Tangent)	tan 2A = (2 tan A)/(1 - tan²A)	Tangent of double angle
Triple Angle (Sine)	sin 3A = 3 sin A - 4 sin³A	Sine of triple angle expansion
Triple Angle (Cosine)	cos 3A = 4 cos³A - 3 cos A	Cosine of triple angle expansion
Sum Formula (Sine)	sin(A + B) = sin A cos B + cos A sin B	Addition formula for sine
Sum Formula (Cosine)	cos(A + B) = cos A cos B - sin A sin B	Addition formula for cosine
Sum Formula (Tangent)	tan(A + B) = (tan A + tan B)/(1 - tan A tan B)	Addition formula for tangent
Product to Sum	2 sin A cos B = sin(A + B) + sin(A - B)	Converts products to sums
Sum to Product	sin A + sin B = 2 sin[(A+B)/2] cos[(A-B)/2]	Converts sums to products

Compound Angles: Advanced Problem-Solving Techniques

Compound angle formulas represent a quantum leap in trigonometric problem-solving capability. When dealing with angles formed by algebraic combinations of simpler angles, these formulas provide systematic approaches to finding exact values. The addition formulas for sine and cosine—sin(A + B) = sin A cos B + cos A sin B and cos(A + B) = cos A cos B - sin A sin B—serve as master keys unlocking solutions to complex trigonometric equations. These relationships extend naturally to subtraction formulas and multiple angle scenarios, creating a comprehensive framework for angle manipulation.

Multiple Angle Formulas and Their Applications

The progression from single angles to multiple angles showcases the elegant scalability of trigonometric functions. Double angle formulas, particularly cos 2A in its three equivalent forms (cos²A - sin²A, 2cos²A - 1, and 1 - 2sin²A), demonstrate the flexibility inherent in trigonometric expressions. Each form serves specific computational advantages, allowing mathematicians to choose the most efficient approach for any given problem. Triple angle formulas further extend this capability, with expressions like sin 3A = 3 sin A - 4 sin³A providing direct routes to otherwise complex calculations.

Special Angles and Non-Standard Values

Beyond the commonly memorized angles of 30°, 45°, and 60°, trigonometry encompasses a rich tapestry of special angles with exact trigonometric values. Angles like 15°, 18°, 36°, and their complementary partners possess precise algebraic expressions involving surds and golden ratio relationships. For instance, sin 18° = (√5 - 1)/4 reveals the unexpected connection between trigonometry and the golden ratio, while cos 15° = (√3 + 1)/(2√2) demonstrates the layered complexity within seemingly simple angle measurements. These special values prove invaluable in competitive examinations and advanced mathematical applications.

Practical Problem-Solving Strategies

Successful navigation of trigonometric problems requires more than memorization—it demands strategic thinking and pattern recognition. When encountering expressions like sin⁶A + cos⁶A + 3sin²A cos²A, recognizing the underlying algebraic identity (a³ + b³ + 3ab formula) transforms a complex expression into the simple result of 1. Similarly, understanding complementary and supplementary angle relationships allows for rapid simplification of expressions involving angles like (90° - θ) or (180° + θ). These problem-solving techniques, combined with a solid grasp of fundamental identities, empower students to tackle increasingly sophisticated mathematical challenges with confidence and precision.

The mastery of trigonometric functions solutions opens doors to advanced mathematics, physics, and engineering applications. From analyzing periodic phenomena to solving complex geometric problems, these mathematical tools provide the foundation for understanding our quantitative world. Regular practice with these formulas and concepts, coupled with strategic problem-solving approaches, ensures not just academic success but also develops the mathematical intuition essential for higher-level scientific thinking.

Definition of Trigonometric Functions

In a right angled triangle ABC, ∠CAB = A and ∠BCA = 90° = π/2. AC is the base, BC the altitude and AB is the hypotenuse. We refer to the base as the adjacent side and to the altitude as the opposite side. There are six trigonometric ratios, also called trigonometric functions or circular functions. With reference to angle A, the six ratios are:

BC/AB = opposite side / hypotenuse is called sine of A, and written as sin A

AC/AB = adjacent side / hypotenuse is called the cosine of A, and written as cos A

BC / AC = opposite side / adjacent side is called the tangent of A, and written as tan A.

Obviously, tan A = sin A/cos A. The reciprocals of sine, cosine and tangent are called the cosecant, secant and cotangent of A respectively. We write these as cosec A, sec A, cot A respectively.

Basic Trigonometric Formulae

Basic Formulas

Trigonometric Ratios of Any Angle

α equals	sin α	cos α	tan α	cot α	Sec α	cosec α
-θ	-sin θ	cos θ	-tan θ	-cot θ	sec θ	-cosec θ
90° - θ	cos θ	sin θ	cot θ	tan θ	cosec θ	sec θ
90° + θ	cos θ	-sin θ	-cot θ	-tan θ	-cosec θ	sec θ
180° - θ	sin θ	-cos θ	-tan θ	-cot θ	-sec θ	cosec θ
180° + θ	-sin θ	-cos θ	tan θ	cot θ	-sec θ	-cosec θ
360° - θ	-sin θ	cos θ	-tan θ	-cot θ	sec θ	-cosec θ
360° + θ	sin θ	cos θ	tan θ	cot θ	sec θ	cosec θ

Notes:

• Angle θ and 90° − θ are complementary angles; θ and 180° − θ are supplementary angles
• sin(nπ + (−1)ⁿθ) = sinθ, n ∈ ℤ
• cos(2nπ ± θ) = cosθ, n ∈ ℤ
• tan(nπ + θ) = tanθ, n ∈ ℤ

Trigonometric Ratios of compounds angles

An angle made up of the algebraic sum of two or more angles is called a compound angle. Some of the formulae and results regarding compound angles are:

• sin(A + B) = sin A cos B + cos A sin B
• sin(A – B) = sin A cos B – cos A sin B
• cos(A + B) = cos A cos B – sin A sin B
• cos(A – B) = cos A cos B + sin A sin B
• tan(A + B) = (tan A + tan B) / (1 – tan A tan B)
• tan(A – B) = (tan A – tan B) / (1 + tan A tan B)
• sin(A + B) sin(A – B) = sin²A – sin²B = cos²B – cos²A
• cos(A + B) cos(A – B) = cos²A – sin²B = cos²B – sin²A

Trigonometric Ratios of Multiples of an angle

• sin2A = 2sinA cosA = ^{2 tanA}_{1 + tan²A}
• cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1 = ^{1 − tan2A}_{1 + tan²A}
• tan2A = ^{2 tanA}_{1 − tan²A}
• sin3A = 3sinA − 4sin³A = 4sin(60° − A) sinA sin(60° + A)
• cos3A = 4cos³A − 3cosA = 4cos(60° − A) cosA cos(60° + A)
• tan3A = ^{3 tanA − tan3A}_{1 − 3 tan²A} = tan(60° − A) tanA tan(60° + A)

SUM OF SINES/COSINES IN TERMS OF PRODUCTS

• sinA + sinB = 2sin[(A+B)/2] cos[(A−B)/2]
• sinA − sinB = 2sin[(A−B)/2] cos[(A+B)/2]
• cosA + cosB = 2cos[(A−B)/2] cos[(A+B)/2]
• cosA − cosB = 2sin[(B−A)/2] sin[(A+B)/2] (here notice (B−A)!)

Conversely:

• 2sinAcosB = sin(A+B) + sin(A−B)
• 2cosAsinB = sin(A+B) − sin(A−B)
• 2cosAcosB = cos(A+B) + cos(A−B)
• 2sinAsinB = cos(A−B) − cos(A+B)

Example: sin 12° · sin 48° · sin 54° is equal to

(A) 1

(B) 1/4

(C) 1/8

(D) none of these

Solution: (C).

Let θ = 12°.
L.H.S. = (1/sin 72°) sin12° sin48° sin72° sin54°
= (1/4) sin 3(12°) sin54° / sin72°
= sin36° sin54° / 8 sin36° cos36°
= cos36° / 8 cos36°
= 1/8 = R.H.S.

Trigonometric Series:

∏_r=0^n-1 cos 2^r A = sin 2ⁿ A
2ⁿ sin A where ∏ denotes products.

Formulas and Concepts

Trigonometric Formulae

Identitities

1. sin²x + cos²x = 1, for all x ∈ ℝ
2. cosec²x = 1 + cot²x, x ≠ nπ
3. sec²x = 1 + tan²x, x, x ≠ (2n + 1)π/2

Sum and Difference Fromula

1. sin(A + B) = sinA cosB + cosA sinB
2. sin(A − B) = sinA cosB − cosA sinB
3. cos(A + B) = cosA cosB − sinA sinB
4. cos(A − B) = cosA cosB + sinA sinB
5. tan(A + B) = (tanA + tanB) / (1 − tanA tanB)
6. tan(A − B) = (tanA − tanB) / (1 + tanA tanB)
7. sin(A + B) sin(A − B) = sin²A − sin²B = cos²B − cos²A
8. cos(A + B) cos(A − B) = cos²A − sin²B = cos²B − sin²A
9. 2sinA cosB = sin(A + B) + sin(A − B)
10. 2cosA cosB = cos(A + B) + cos(A − B)

Multiple Angle Formulae

1. sin2A = 2sinA cosA = (2 tanA) / (1 + tan²A)
2. cos2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1 = (1 − tan²A) / (1 + tan²A)
3. sin²A = (1 − cos2A)/2 and cos²A = (1 + cos2A)/2
4. tan2A = (2 tanA) / (1 − tan²A)
5. tan3A = (3 tanA − tan³A) / (1 − 3 tan²A)
6. sin3A = 3sinA − 4sin³A ⇒ sin³A = (3 sinA − sin3A) / 4
7. cos3A = 4cos³A − 3cosA ⇒ cos³A = (3 cosA + cos3A) / 4

Trigonometric Ratios of some non-standard angles

1. sin15° = cos75° = (√3 − 1) / (2√2)
2. cos15° = sin75° = (√3 + 1) / (2√2)
3. sin18° = cos72° = (√5 − 1) / 4
4. sin54° = cos36° = (√5 + 1) / 4
5. sin36° = cos54° = (√10 − 2√5) / 4
6. cos18° = sin72° = (√10 + 2√5) / 4
7. sin22.5° = ½√(2 − √2)
8. cos22.5° = ½√(2 + √2)
9. tan18° = 2 − √3
10. tan22.5° = √2 − 1

Solved Examples

Ex.: If sin θ + cosec θ = 2, then sin² θ + cosec² θ is equal to

(A) 1

(B) 4

(C) 2

(D) none of these

Sol. (C). Given sin θ + cosec θ = 2 ; ⇒ (sin θ + cosec θ)² = 4
⇒ sin² θ + cosec² θ + 2 = 4 ; ⇒ sin² θ + cosec² θ = 2.

Ex.: If (1 + tan θ)(1 + tan ϕ) = 2, then θ + ϕ is equal to

(A) 30°

(B) 45°

(C) 60°

(D) 75°

Sol. Given (1 + tan θ)(1 + tan ϕ) = 2 ; ⇒ 1 + tan θ + tan ϕ + tan θ tan ϕ = 2
⇒ tan θ + tan ϕ = 1 – tan θ tan ϕ ;

Ex.: If x = y cos(2π/3) = z cos(4π/3), then xy + yz + zx is equal to

(A) –1

(B) 0

(C) 1

(D) 2

Sol. (B). Given x = y cos(2π/3) = z cos(4π/3); ⇒ x = y (–1/2) = z (–1/2)
⇒ x/1 = y/(–2) = z/(–2) = λ (say); ⇒ x = λ, y = –2λ, z = –2λ.
Hence xy + yz + zx = λ(–2λ) + (–2λ)(–2λ) + (–2λ)λ = 0.

Ex.: The value of cos(2π/7) + cos(4π/7) + cos(6π/7) + cos(7π/7) is equal to

(A) 1

(B) –1

(C) 1/2

(D) –3/2

Sol. (D). cos(2π/7) + cos(4π/7) + cos(6π/7) + cos(7π/7)
= 1/(2 sin(π/7)) {2 cos(2π/7) sin(π/7) + 2 cos(4π/7) sin(π/7) + 2 cos(6π/7) sin(π/7)} + cos π
= –1/2 – 1 = –3/2