Sine (sin θ)
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Trigonometry Cheat Sheet
A comprehensive cheat sheet covering essential trigonometry concepts, identities, functions, and formulas. Useful for quick reference and exam preparation.
Basic Trigonometric Functions
Right Triangle Definitions
|
|
opposite / hypotenuse |
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Cosine (cos θ) |
adjacent / hypotenuse |
|
Tangent (tan θ) |
opposite / adjacent |
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Cosecant (csc θ) |
hypotenuse / opposite (1 / sin θ) |
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Secant (sec θ) |
hypotenuse / adjacent (1 / cos θ) |
|
Cotangent (cot θ) |
adjacent / opposite (1 / tan θ) |
Reciprocal Identities
|
sin θ |
1 / csc θ |
|
cos θ |
1 / sec θ |
|
tan θ |
1 / cot θ |
|
csc θ |
1 / sin θ |
|
sec θ |
1 / cos θ |
|
cot θ |
1 / tan θ |
Quotient Identities
|
tan θ |
sin θ / cos θ |
|
cot θ |
cos θ / sin θ |
Trigonometric Identities
Pythagorean Identities
|
sin² θ + cos² θ = 1 |
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1 + tan² θ = sec² θ |
|
1 + cot² θ = csc² θ |
Angle Sum and Difference Identities
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sin(A + B) |
sin A cos B + cos A sin B |
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sin(A - B) |
sin A cos B - cos A sin B |
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cos(A + B) |
cos A cos B - sin A sin B |
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cos(A - B) |
cos A cos B + sin A sin B |
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tan(A + B) |
(tan A + tan B) / (1 - tan A tan 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 = 2 cos² θ - 1 = 1 - 2 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) |
sin θ / (1 + cos θ) = (1 - cos θ) / sin θ |
Unit Circle
Common Angles
|
0° (0 radians) |
(1, 0) |
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30° (π/6 radians) |
(√3/2, 1/2) |
|
45° (π/4 radians) |
(√2/2, √2/2) |
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60° (π/3 radians) |
(1/2, √3/2) |
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90° (π/2 radians) |
(0, 1) |
|
180° (π radians) |
(-1, 0) |
|
270° (3π/2 radians) |
(0, -1) |
Sine and Cosine Values
|
sin θ |
y-coordinate |
|
cos θ |
x-coordinate |
Tangent Values
|
tan θ = sin θ / cos θ = y / x |
Laws and Formulas
Law of Sines
|
a / sin A = b / sin B = c / sin C |
Law of Cosines
|
a² = b² + c² - 2bc cos A |
|
b² = a² + c² - 2ac cos B |
|
c² = a² + b² - 2ab cos C |
Area of a Triangle
|
Area (using sides and angle) |
1/2 * ab * sin C = 1/2 * bc * sin A = 1/2 * ac * sin B |
|
Heron’s Formula (using sides) |
Area = √(s(s - a)(s - b)(s - c)), where s = (a + b + c) / 2 |