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Trigonometry With Calculus

The document provides a comprehensive overview of trigonometric concepts, including basic ratios, identities, formulas, and their applications in calculus. It covers various identities such as Pythagorean, negative angle, cofunction, and sum/difference formulas, along with derivatives and integrals of trigonometric functions. Additionally, it includes standard angle values and the unit circle representation.

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89 views5 pages

Trigonometry With Calculus

The document provides a comprehensive overview of trigonometric concepts, including basic ratios, identities, formulas, and their applications in calculus. It covers various identities such as Pythagorean, negative angle, cofunction, and sum/difference formulas, along with derivatives and integrals of trigonometric functions. Additionally, it includes standard angle values and the unit circle representation.

Uploaded by

sumedh7sss
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Comprehensive Trigonometry with Calculus (Up to

Undergraduate Level)

Basic Trigonometric Ratios


sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent = sin θ / cos θ
cot θ = 1 / tan θ = cos θ / sin θ
sec θ = 1 / cos θ
csc θ = 1 / sin θ

Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Negative Angle Identities


sin(-θ) = -sin θ
cos(-θ) = cos θ
tan(-θ) = -tan θ
cot(-θ) = -cot θ
sec(-θ) = sec θ
csc(-θ) = -csc θ

Cofunction Identities
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

Sum and Difference Formulas


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 Formulas


sin(2A) = 2 sin A cos A
cos(2A) = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
tan(2A) = 2tan A / (1 - tan²A)
Half Angle Formulas
sin²(A/2) = (1 - cos A) / 2
cos²(A/2) = (1 + cos A) / 2
tan²(A/2) = (1 - cos A) / (1 + cos A)

Product to Sum Formulas


sin A sin B = ½[cos(A - B) - cos(A + B)]
cos A cos B = ½[cos(A - B) + cos(A + B)]
sin A cos B = ½[sin(A + B) + sin(A - B)]

Sum to Product Formulas


sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2)
sin A - sin B = 2 cos((A + B)/2) sin((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)

Law of Sines, Cosines, Tangents


Law of Sines: a/sin A = b/sin B = c/sin C = 2R
Law of Cosines: c² = a² + b² - 2ab cos C
Law of Tangents: (a - b)/(a + b) = tan((A - B)/2) / tan((A + B)/2)

Inverse Trigonometric Identities


sin(sin■¹x) = x, for -1 ≤ x ≤ 1
cos(cos■¹x) = x, for -1 ≤ x ≤ 1
tan(tan■¹x) = x, for all real x
sin■¹x + cos■¹x = π/2
tan■¹x + cot■¹x = π/2
sec■¹x + csc■¹x = π/2

Miscellaneous Identities
sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2), when A + B + C = π
cos A + cos B + cos C = 1 + r/R (for triangle)

Standard Angle Values


Angle sin θ cos θ tan θ
0° 0 1 0
30° 1/2 √3/2 1/√3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
90° 1 0 ∞
Unit Circle

Sine Function y = sin(x)

Cosine Function y = cos(x)


Tangent Function y = tan(x_vals)

Calculus with Trigonometric Functions

Derivatives of Trigonometric Functions


d/dx [sin x] = cos x
d/dx [cos x] = -sin x
d/dx [tan x] = sec²x
d/dx [cot x] = -csc²x
d/dx [sec x] = sec x · tan x
d/dx [csc x] = -csc x · cot x
d/dx [sin■¹x] = 1 / √(1 - x²), |x| < 1
d/dx [cos■¹x] = -1 / √(1 - x²), |x| < 1
d/dx [tan■¹x] = 1 / (1 + x²)
Integrals of Trigonometric Functions
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ csc²x dx = -cot x + C
∫ sec x tan x dx = sec x + C
∫ csc x cot x dx = -csc x + C
∫ dx / (1 + x²) = tan■¹x + C
∫ dx / √(1 - x²) = sin■¹x + C
∫ -dx / √(1 - x²) = cos■¹x + C

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