Trigonometric Identities and T- ratios of Allied Angles | Physics for JEE Main & Advanced PDF Download

Basic Trigonometric Identities

• Basic trigonometric identities are fundamental equations that relate the angles and sides of a right triangle. They help us understand the relationships between trigonometric functions like sine, cosine, and tangent. 
• Understanding these identities is crucial for solving trigonometric equations and problems in geometry and physics. They form the building blocks for more complex trigonometric manipulations and calculations.

Following are the most common trigonometric Identities:

(i) sin θ. cosec θ = 1
(ii) cos θ. sec θ = 1
(iii) tan θ. cot θ = 1
(iv) tan θ = sin θ / cos θ & cot θ = cos θ / sin θ
(v) sin² θ + cos² θ = 1  or sin² θ = 1 - cos² θ  or cos² θ = 1 - sin² θ
(vi) sec² θ - tan² θ = 1  or sec² θ = 1 + tan² θ  or tan² θ = sec² θ  -  1
(vii) 
(viii) cosec² θ - cot² θ = 1  or  cosec² θ = 1 + cot² θ or  cot² θ  = cosec² θ - 1
(ix) 
(x) Expressing trigonometrical ratio in terms of each other:

• Expressing trigonometric ratios in terms of each other means finding relationships between sine, cosine, tangent, and other trigonometric functions. For example, we can express tangent as sine divided by cosine, or cosine as sine divided by tangent. 
• These relationships help simplify trigonometric expressions and solve trigonometric equations more efficiently.

Trigonometrical Ratio in Terms of Each Other

Example 1: If sin θ + sin² θ = 1 , then prove that cos¹² θ + 3 cos¹⁰ θ + 3 cos⁸ θ + cos⁶ θ - 1 = 0
Sol: Given that sin θ = 1 - sin² θ = cos² θ
L.H.S. = cos⁶ θ (cos² θ + 1)³ - 1= sin³ θ (1 + sin θ )³ - 1= (sin θ + sin² θ)³ - 1 = 1 - 1 = 0

Example 2: (sin⁶ θ + cos⁶ θ) - 3 ( sin⁴ θ + cos⁴ θ) + 1 is equal to
(a) 0
(b) 1
(c) –2
(d) none of these
Ans: (a)
2 [(sin² θ + cos² θ )³ - 3 sin² θ cos² θ ( sin² θ + cos² θ) ]  - 3 [(sin² θ + cos² θ)]² - 2sin² θ cos² θ] + 1
= 2 [1 – 3 sin² θ cos² θ] - 3 [1 - 2 sin² θ cos² θ] + 1
= 2 - 6 sin2 θ cos² θ - 3 + 6 sin² θ cos² θ + 1 = 0

Signs of Trigonometric Functions in Different Quadrants

Understanding the signs of trigonometric functions in different quadrants is crucial in trigonometry. 

• In the first quadrant, all trigonometric functions are positive because both the x and y coordinates are positive. 
• In the second quadrant, only sine is positive, while cosine and tangent are negative. 
• In the third quadrant, only tangent is positive, while sine and cosine are negative. 
• In the fourth quadrant, only cosine is positive, while sine and tangent are negative. 

These signs help determine the positive or negative values of trigonometric functions based on the angle's location in the coordinate plane.

Signs of Trigonometric Ratios in Different Quadrants

Trigonometric Functions of Allied Angles

• Trigonometric functions of allied angles refer to angles that differ by multiples of 90 degrees (or π/2 radians). 
• For example, 30 degrees and 120 degrees are allied angles because they sum up to 150 degrees, which is 90 degrees plus 60 degrees. 
• Understanding the relationships between trigonometric functions of such angles helps simplify trigonometric expressions and solve problems more efficiently. 
• Following are some of the common trigonometric functions of allied angles.
  Some Common Trigonometric Functions of Allied Angles

Values of T-ratios of Some Standard Angles

• Values of T-ratios (trigonometric ratios) of some standard angles refer to the specific values of sine, cosine, and tangent for commonly used angles like 0°, 30°, 45°, 60°, and 90°. 
• These values are often memorized or calculated and are fundamental in trigonometry. 
• For instance, the sine of 30° is 1/2, cosine of 45° is √2/2, and tangent of 60° is √3. 
• Knowing these standard values helps quickly solve trigonometric problems and understand the behavior of trigonometric functions for different angles.
• The following table depicts the values of T-ratios of some standard angles.

Values of T-ratios of Some Standard AnglesN.D. → Not Defined
(a) sin nπ = 0 ; cos nπ = (-1)ⁿ; tan nπ = 0 where n ∈ I
(b) sin(2n + 1) π / 2 = (-1)ⁿ; cos(2n + 1) π / 2 = 0 where n ∈ I

Example 3: If sin θ = -1 / 2 and tan θ = 1 / √3 then θ is equal to:
(a) 30°
(b) 150°
(c) 210°
(d) None of these
Ans: (c)
Let us first find out θ lying between 0 and 360°.
Since sin θ = -1 / 2 ⇒ θ = 210° or 330° and tan θ = 1 / √3 ⇒ θ = 30° or 210°
Hence, θ = 210° or 7π / 6 is the value satisfying both.