CBSE 10 Trigonometry: How to Master the Mystical Identities (Complete List of Formulas and Equations)

We understand the significance of Trigonometric Identities in the world of trigonometry. These identities play a crucial role in simplifying and manipulating trigonometric expressions, enabling us to solve complex problems with ease. In this article, we will explore various Trigonometric Identities, their applications, and solve example questions to deepen your understanding.

What are trigonometric identities?

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for all values of the variables involved. They are derived from the fundamental relationships between the sides and angles of a right triangle. These identities serve as valuable tools to simplify trigonometric expressions, making calculations more manageable.

List of trigonometric identities

Reciprocal identities

• csc θ = 1 / sin θ
• sec θ = 1 / cos θ
• cot θ = 1 / tan θ

Pythagorean identities

• sin² θ + cos² θ = 1
• tan² θ + 1 = sec² θ
• 1 + cot² θ = csc² θ

Ratio identities

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent
• cot θ = adjacent / opposite

Identities of opposite angles

• sin (-θ) = -sin θ
• cos (-θ) = cos θ
• tan (-θ) = -tan θ

Identities of complementary angles

• sin (90° - θ) = cos θ
• cos (90° - θ) = sin θ
• tan (90° - θ) = cot θ
• cot (90° - θ) = tan θ
• sec (90° - θ) = csc θ
• csc (90° - θ) = sec θ

Identities of supplementary angles

• sin (180° - θ) = sin θ
• cos (180° - θ) = -cos θ
• tan (180° - θ) = -tan θ

Sum and difference of angles identities

• sin (A ± B) = sin A * cos B ± cos 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)

Double angle identities

• sin (2θ) = 2 * sin θ * cos θ
• cos (2θ) = cos² θ - sin² θ
• tan (2θ) = (2 tan θ) / (1 - tan² θ)

Half angle identities

• sin (θ/2) = ±√[(1 - cos θ) / 2]
• cos (θ/2) = ± √[(1 + cos θ ) / 2]
• tan (θ/2) = ± √[(1 - cos θ) / (1 + cos θ)]

Sum-to-product identities

• sin A + sin B = 2 * sin [(A + B) / 2] * cos [(A - B) / 2]
• sin A - sin B = 2 * sin [(A - B) / 2] * cos [(A + B) / 2]
• cos A + cos B = 2 * cos [(A + B)/ 2] * cos [(A - B) / 2]
• cos A - cos B = -2 * sin [(A + B) / 2] * sin [(A - B) / 2]

Product-to-sum trigonometric identities

• sin A * sin B = (1/2) * [cos (A - B) - cos(A + B)]
• cos A * cos B = (1/2) * [cos (A - B) + cos (A + B)]
• sin A * cos B = (1/2) * [sin (A + B) + sin (A - B)]
• cos A * sin B = (1/2)* [sin (A + B) - sin (A - B)]

Triangle identities (sine, cosine, and tangent Rules)

• The Sine Rule: a/sin A = b/sin B = c/sin C, where a, b, and c are the the sides of the triangle while A, B, and C are their opposite angles, respectively
• The Cosine Rule: c² = a² + b² - 2 * ab * cos C, where a, b, and c are the the sides of the triangle while A, B, and C are their opposite angles, respectively

Trigonometric identities questions

Example 1:

Simplify the expression (sin² θ - cos² θ) / (tan θ - cot θ).

✅  Solution:
      1. Apply the Ratio Identities: tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.
      2. Substitute the identities into the expression:
          (sin² θ - cos² θ) / [(sin θ / cos θ) - (cos θ / sin θ )].
      3. Simplify the denominator: (sin² θ - cos² θ) / (sin θ * cos θ).
      4. Substitute the denominator into the expression:
          (sin² θ - cos² θ) / [(sin² θ - cos² θ) / (sin θ * cos θ)].
      5. Simplify the expression: sin θ * cos θ.

Example 2:

Find the exact value of cos 105°.

‍✅  Solution:
      1. Notice the fact: 105° = 45° + 60°.
      2. Apply the Sum of Angles Identity for cosines:
          cos (45° + 60°) = cos 45° * cos 60° - sin 45° * sin 60°.
      3. Recall the facts: cos 45° = √2/2, cos 60° = 1/2, sin 45° = √2/2, and sin 60° = √3/2.
      4. Substitute the values: cos 105° = (√2/2) * (1/2) - (√2/2) * (√3/2).
      5. Simplify: cos 105° = (√2 - √6) / 4.

Trigonometric functions questions

Example 3:

Find the exact value of sin (π/4).

‍✅  Solution:
      1. Notice the fact: π/4 can be represented as 45°.
      2. Apply the Ratio Identity: sin θ = opposite / hypotenuse.
      3. In a 45° right-angled triangle, the opposite and adjacent sides are equal.
      4. Since the hypotenuse is √2 times the length of either the opposite or adjacent
          side, sin (π/4) = 1/√2.

FAQs

What are the three main functions in trigonometry?

The three main trigonometric functions are sine (sin), cosine (cos), and tangent (tan).

What are the basic 8 trigonometric identities?

The basic 8 trigonometric identities include three reciprocal identities, three Pythagorean identities, and two ratio identities.

What are the Pythagoras identities?

The Pythagorean identities are sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, and 1 + cot² θ = csc² θ.

What is the reciprocal of the sine function?

The reciprocal of the sine function is the cosecant function, given by csc θ = 1 / sin θ.

What is the value of sin 2A?

The value of sin 2A can be represented as 2 * sin A * cos A.

Conclusion

In conclusion, Trigonometric Identities are fundamental tools in trigonometry that help simplify expressions and solve complex problems. By understanding these identities and their applications, you will enhance your ability to tackle various trigonometric calculations and challenges.

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Relevant links

• CBSE 10 - Mensuration - Areas Related to Circles
• Trigonometric Equations and Identities by Khan Academy
• Trigonometric Identities by Math is Fun
• Trigonometric Identities by BYJU'S