11 Questions
What is the range of the sine function?
[-1, 1]
What is the period of the sine function in radians?
2π
What is the Pythagorean identity related to the sine function?
sin^2(x) + cos^2(x) = 1
What is the sum formula for the sine function?
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
What is the inverse of the sine function denoted by?
arcsin(x) or sin^(-1)(x)
What is one of the applications of the sine function in physics?
Modeling periodic phenomena, such as sound waves and light waves
Wat is de définitie van de sinusfunctie in een rechtshoekig driehoek?
De sinusfunctie wordt gedefinieerd als de verhouding van de overstaande zijde tot de hypotenusa van een rechtshoekig driehoek.
Waarom is de cosinusfunctie een even functie?
De cosinusfunctie is een even functie omdat cos(-A) = cos(A).
Wat is de relatie tussen de sinus en cosinusfunctie in een rechtshoekig driehoek?
De relatie tussen de sinus en cosinusfunctie is gegeven door dePythagoreïsche identiteit: sin²(A) + cos²(A) = 1.
Hoe kan men de sinusfunctie van een hoek A berekenen met behulp van de cosinusfunctie?
Men kan de sinusfunctie van een hoek A berekenen met behulp van de cosinusfunctie door gebruik te maken van de relatie: sin(A) = cos(π/2 - A).
Wat is de waarde van sin(A) + cos(A) voor een willekeurige hoek A?
Er is geen algemene waarde voor sin(A) + cos(A) voor een willekeurige hoek A, omdat deze waarde afhankelijk is van de specifieke hoek A.
Study Notes
Sine
Definition: The sine function is a trigonometric function that relates the ratio of the opposite side to the hypotenuse of a right-angled triangle.
Notation: The sine function is denoted by sin(x), where x is the angle in radians or degrees.
Unit Circle: The sine function can be defined using the unit circle, where the sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.
Properties:
- Range: The range of the sine function is [-1, 1].
- Period: The period of the sine function is 2π (or 360°).
- Odd Function: The sine function is an odd function, meaning sin(-x) = -sin(x).
- Inverse: The inverse of the sine function is denoted by arcsin(x) or sin^(-1)(x).
Identities:
- Pythagorean Identity: sin^2(x) + cos^2(x) = 1
-
Sum and Difference Formulas:
- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
- sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Graph: The graph of the sine function is a wave-like curve with a maximum value of 1 and a minimum value of -1.
Applications: The sine function has many applications in physics, engineering, and mathematics, including:
- Modeling periodic phenomena, such as sound waves and light waves
- Calculating distances and angles in trigonometry
- Analyzing circular motion and oscillations
Sine Function
- Relates the ratio of the opposite side to the hypotenuse of a right-angled triangle.
- Denoted by sin(x), where x is the angle in radians or degrees.
Unit Circle Definition
- Sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.
Properties
- Range: [-1, 1]
- Period: 2π (or 360°)
- Odd function: sin(-x) = -sin(x)
- Inverse: arcsin(x) or sin^(-1)(x)
Trigonometric Identities
- Pythagorean Identity: sin^2(x) + cos^2(x) = 1
- Sum and Difference Formulas:
- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
- sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Graph Characteristics
- Wave-like curve with a maximum value of 1 and a minimum value of -1
Applications
- Modeling periodic phenomena, such as sound waves and light waves
- Calculating distances and angles in trigonometry
- Analyzing circular motion and oscillations
Trigonometric Functions
Sine (sin)
- Defined as the ratio of the opposite side to the hypotenuse of a right-angled triangle
- sin(A) equals opposite side divided by hypotenuse
- Range is between -1 and 1 (inclusive)
- Period is 2π, meaning the function repeats every 2π radians
- Odd function, meaning sin(-A) is equal to -sin(A)
Cosine (cos)
- Defined as the ratio of the adjacent side to the hypotenuse of a right-angled triangle
- cos(A) equals adjacent side divided by hypotenuse
- Range is between -1 and 1 (inclusive)
- Period is 2π, meaning the function repeats every 2π radians
- Even function, meaning cos(-A) is equal to cos(A)
Relationships between Sine and Cosine
- Pythagorean identity: sin²(A) + cos²(A) equals 1
- sin(A) equals cos(π/2 - A)
- cos(A) equals sin(π/2 - A)
Learn about the sine function, its notation, and properties in the context of right-angled triangles and the unit circle.
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