Analytic trigonometry uses algebraic and geometric methods to solve problems with trigonometric functions(1).pptx

Analytic trigonometry uses algebraic and geometric methods to solve problems with trigonometric functions, building on basic trigonometry by using a coordinate system to analyze functions and their properties. It focuses on manipulating trigonometric identities and formulas to simplify expressions, solve equations, and prove relationships between functions, which are crucial for applications in calculus, engineering, and other sciences. Key topics include proving identities, sum and difference formulas, double and half-angle formulas, and solving trigonometric equations.

1. Analytic Trigonomertry

2. The Eight Fundamental Identities The fundamental identities are grouped into three-the Reciprocal identities , the Pythagorean identities and the Ratio or Quotient identities .

3. From the definition of the trigonometric functions, Reciprocal Identity 1 - Reciprocal identity refers to the relationship between two or more entities that are interconnected and interdependent, where each entity's identity is influenced by and reflected in the other. 1 If we multiply the two functions,

4. Reciprocal Identity 1 Similarly, so, and, so,

5. Pythagorean Identities 1 Pythagorean Identities are important identities in trigonometry that are derived from the Pythagoras theorem . These identities are used in solving many trigonometric problems where one trigonometric ratio is given and the other ratios are to be found. The fundamental Pythagorean identity gives the relation between sin and cos and it is the most commonly used Pythagorean identity which says: 1 (which gives the relation between sin and cos) There are other two Pythagorean identities that are as follows: (which gives the relation between sec and tan ) (which gives the relation between csc and cot )

6. Quotient Identities 1 Quotient identities are fundamental trigonometric identities that relate the tangent and cotangent functions to the sine and cosine functions. Specifically, they define the ratio (quotient) of sine to cosine for tangent, and cosine to sine for cotangent. 1 Tangent Identity: tan⁡ θ = sin ⁡θ/cos ⁡θ. Cotangent Identity: cot⁡ θ = cos⁡ θ/sin⁡ θ.

7. Proving Identities

8. We have learned how to express a trigonometric expression in different forms. In proving an identity, we must show that the left and right sides of the given equation are just different forms of the same expression. There is no best procedure in proving that an equation is an identity, but it is good to work on the more complicated side so that the application of algebraic operations can reduce it to the other simpler form

9. We may work with the left side only, or with the right side only, or with both in order to obtain exactly the same expression. Caution: In proving identities, strictly: NO transposition of terms from one side to the other, and NO cross multiplication since these two processes will change the value of each side of the equation.

10. Example : Prove that is an identity. Solution : Since the left side is more complicated, we perform algebraic operations on it.

11. The Sum and Difference Identities

12. The sum and difference identities help find the trigonometric values of non-special angles using the known trigonometric values of special angles, derived from the unit circle. Sum identities are used when two special angles are added to get the non-special angle, and difference identities are used when they are subtracted.

13. Sum and Difference Formulas cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B tan(A + B) = (tanA + tanB) / (1 - tanA tanB) tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

14. Example : Find the values of sin 75° and tan 75° using the sum and difference identities. Solution : We can write 75° as 75° = 45° + 30°. So, using the sum formulas of sine and tangent, we have sin 75° = sin (45° + 30°) = sin 45° cos 30° + sin 30° cos 45° --- [Using sin(A + B) = sinA cosB + cosA sinB] = (1/√2) (√3/2) + (1/2) (1/√2) --- [Because sin 45° = 1/√2, cos 30° = √3/2, sin 30° = 1/2, and cos 45° = 1/√2]

15. = √3/2√2 + 1/2√2 = (√3 + 1) / 2√2 Similarly, we can calculate the value of tan 75°. tan 75° = tan (45° + 30°) = (tan45° + tan30°) / (1 - tan45° tan30°) --- [Using tan(A + B) = (tanA + tanB) / (1 - tanA tanB)] = (1 + 1/√3) / (1 - 1 × 1/√3) --- [Because tan 45° = 1, and tan 30° = 1/√3] = (√3 + 1) / (√3 - 1) Answer: sin 75° = (√3 + 1) / 2√2 and tan 75° = (√3 + 1) / (√3 - 1) v

16. Double Angle and Half Angle Identities

17. Double Angle Identities 1 - The double angle identities are proved by applying the sum and difference identities. They are left as review problems. These are the double angle identities. 1

18. Half Angle Identities 1 - The half angle identities are a rewritten version of the power reducing identities. The proofs are left as review problems. 1

19. Any Questions? Thank you for listening