Questions and Answers
What is the definition of sine in a right triangle?
What is the sum formula for sine in trigonometry?
What is the Pythagorean Identity in trigonometry?
What is the application of trigonometry in navigation?
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What type of triangle is not a right triangle?
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What is the unit of measurement for angles in radians?
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What is the trigonometric ratio of the adjacent side to the hypotenuse in a right triangle?
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What is the application of trigonometry in modeling periodic phenomena?
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What is the reciprocal of the sine function?
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What is the period of the tangent function?
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What is the formula for the cosine of a double angle?
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What is the identity for sin(A + B) in terms of sine and cosine of A and B?
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What is the trigonometric function that models wave motion?
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What is the trigonometric identity for sin(-A)?
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What is the type of triangle used to define trigonometric ratios?
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What is the value of sin(45°) in a unit circle?
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What is the formula for the tangent of a double angle?
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What is the amplitude of a trigonometric function?
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Study Notes
Trigonometry
Definition
- Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Key Concepts
Angles
- Degrees: A unit of measurement for angles, with 360 degrees in a full circle.
- Radians: A unit of measurement for angles, with 2π radians in a full circle.
Triangles
- Right Triangle: A triangle with one right angle (90 degrees).
- Oblique Triangle: A triangle that is not a right triangle.
Trigonometric Ratios
- Sine (sin): The ratio of the opposite side to the hypotenuse in a right triangle.
- Cosine (cos): The ratio of the adjacent side to the hypotenuse in a right triangle.
- Tangent (tan): The ratio of the opposite side to the adjacent side in a right triangle.
- Cotangent (cot): The ratio of the adjacent side to the opposite side in a right triangle.
- Secant (sec): The ratio of the hypotenuse to the adjacent side in a right triangle.
- Cosecant (csc): The ratio of the hypotenuse to the opposite side in a right triangle.
Identities
- Pythagorean Identity: sin^2(A) + cos^2(A) = 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)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Applications
- Triangulation: Solving triangles using trigonometric ratios and identities.
- Wave Motion: Modeling periodic phenomena, such as sound and light waves, using trigonometric functions.
- Navigation: Calculating distances and directions using trigonometric principles.
Trigonometry
Definition
- Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
Angles
- A full circle has 360 degrees or 2π radians.
- Degrees and radians are units of measurement for angles.
Triangles
- A right triangle has one right angle (90 degrees).
- An oblique triangle is a triangle that is not a right triangle.
Trigonometric Ratios
- Sine (sin) is the ratio of the opposite side to the hypotenuse in a right triangle.
- Cosine (cos) is the ratio of the adjacent side to the hypotenuse in a right triangle.
- Tangent (tan) is the ratio of the opposite side to the adjacent side in a right triangle.
- Cotangent (cot) is the ratio of the adjacent side to the opposite side in a right triangle.
- Secant (sec) is the ratio of the hypotenuse to the adjacent side in a right triangle.
- Cosecant (csc) is the ratio of the hypotenuse to the opposite side in a right triangle.
Identities
- The Pythagorean Identity states that sin^2(A) + cos^2(A) = 1.
- Sum and Difference Formulas include:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Applications
- Triangulation involves solving triangles using trigonometric ratios and identities.
- Wave motion can be modeled using trigonometric functions to represent periodic phenomena, such as sound and light waves.
- Trigonometry has navigation applications, including calculating distances and directions.
Trigonometry
Definitions and Concepts
- An angle is a measure of rotation between two lines or planes
- A triangle is a polygon with three sides and three angles
- A right triangle is a triangle with one right angle (90 degrees)
- The hypotenuse is the side opposite the right angle in a right triangle
- Trigonometric ratios are relationships between the sides and angles of a right triangle
Trigonometric Ratios
- Sine (sin) is the ratio of the opposite side to the hypotenuse
- Cosine (cos) is the ratio of the adjacent side to the hypotenuse
- Tangent (tan) is the ratio of the opposite side to the adjacent side
- Cotangent (cot) is the ratio of the adjacent side to the opposite side
- Secant (sec) is the ratio of the hypotenuse to the opposite side
- Cosecant (csc) is the ratio of the hypotenuse to the adjacent side
Identities and Formulas
- The Pythagorean Identity states that sin^2(A) + cos^2(A) = 1
- The Sum and Difference Formulas are:
- 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))
- The Double Angle Formulas are:
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos^2(A) - sin^2(A)
- tan(2A) = 2tan(A) / (1 - tan^2(A))
Applications and Graphs
- Right triangle problems involve solving for sides and angles using trigonometric ratios
- Wave motion can be modeled using sine and cosine functions
-
Graphs of trigonometric functions have:
- A period of 2π for sine and cosine
- A period of π for tangent
- Amplitude and phase shifts can be applied to trigonometric functions
Important Values and Identities
- The unit circle is a circle with a radius of 1, used to define trigonometric ratios
- Special angles are 30°, 45°, 60°, and their corresponding trigonometric ratios
- Trigonometric identities are equations that are true for all angles, such as sin(-A) = -sin(A)
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Description
Learn about the fundamental concepts of trigonometry, including angles, triangles, and their relationships. Covers degrees, radians, right triangles, and oblique triangles.
