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Questions and Answers
What type of angle is greater than 90 degrees but less than 180 degrees?
What type of angle is greater than 90 degrees but less than 180 degrees?
What is the sum of the interior angles of a triangle?
What is the sum of the interior angles of a triangle?
What is the ratio of the opposite side to the hypotenuse in a right-angled triangle?
What is the ratio of the opposite side to the hypotenuse in a right-angled triangle?
What is the period of the sine wave?
What is the period of the sine wave?
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What is the range of the cosine function?
What is the range of the cosine function?
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What is the ratio of the opposite side to the adjacent side in a right-angled triangle?
What is the ratio of the opposite side to the adjacent side in a right-angled triangle?
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What type of triangle has one angle that is 90 degrees?
What type of triangle has one angle that is 90 degrees?
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What is the period of the tangent wave?
What is the period of the tangent wave?
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Study Notes
Triangles
- A triangle is a polygon with three sides and three angles.
- Types of triangles:
- Right-angled triangle: one angle is 90 degrees (π/2 radians)
- Oblique triangle: no right angles
- Acute triangle: all angles are acute (less than 90 degrees)
- Obtuse triangle: one angle is obtuse (greater than 90 degrees)
- Properties:
- Angle sum: interior angles add up to 180 degrees (π radians)
- Side lengths: can be calculated using trigonometric ratios
Angles
- Measured in degrees (°) or radians (rad)
- Types of angles:
- Acute angle: less than 90 degrees (π/2 radians)
- Right angle: exactly 90 degrees (π/2 radians)
- Obtuse angle: greater than 90 degrees (π/2 radians) but less than 180 degrees (π radians)
- Straight angle: exactly 180 degrees (π radians)
- Angle relationships:
- Complementary angles: add up to 90 degrees (π/2 radians)
- Supplementary angles: add up to 180 degrees (π radians)
Sine (sin)
- Defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle
- sin(A) = opposite side / hypotenuse
- Range: -1 ≤ sin(A) ≤ 1
- Graph: sine wave with period 2π
Cosine (cos)
- Defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle
- cos(A) = adjacent side / hypotenuse
- Range: -1 ≤ cos(A) ≤ 1
- Graph: cosine wave with period 2π
Tangent (tan)
- Defined as the ratio of the opposite side to the adjacent side in a right-angled triangle
- tan(A) = opposite side / adjacent side
- Range: all real numbers
- Graph: tangent wave with period π
Note: These notes provide a concise overview of the key concepts and formulas in trigonometry, focusing on triangles, angles, and the three main trigonometric ratios: sine, cosine, and tangent.
Triangles
- A triangle has three sides and three angles.
Types of Triangles
- A right-angled triangle has one angle of 90 degrees (π/2 radians).
- An oblique triangle has no right angles.
- An acute triangle has all angles less than 90 degrees.
- An obtuse triangle has one angle greater than 90 degrees.
Properties of Triangles
- The interior angles of a triangle add up to 180 degrees (π radians).
- Side lengths of a triangle can be calculated using trigonometric ratios.
Angles
- Angles are measured in degrees (°) or radians (rad).
Types of Angles
- An acute angle is less than 90 degrees (π/2 radians).
- A right angle is exactly 90 degrees (π/2 radians).
- An obtuse angle is greater than 90 degrees (π/2 radians) but less than 180 degrees (π radians).
- A straight angle is exactly 180 degrees (π radians).
Angle Relationships
- Complementary angles add up to 90 degrees (π/2 radians).
- Supplementary angles add up to 180 degrees (π radians).
Sine (sin)
- Sine is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
- The formula for sine is sin(A) = opposite side / hypotenuse.
- The range of sine is -1 ≤ sin(A) ≤ 1.
- The graph of sine is a sine wave with a period of 2π.
Cosine (cos)
- Cosine is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
- The formula for cosine is cos(A) = adjacent side / hypotenuse.
- The range of cosine is -1 ≤ cos(A) ≤ 1.
- The graph of cosine is a cosine wave with a period of 2π.
Tangent (tan)
- Tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle.
- The formula for tangent is tan(A) = opposite side / adjacent side.
- The range of tangent is all real numbers.
- The graph of tangent is a tangent wave with a period of π.
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Description
Learn about the types and properties of triangles, including right-angled, oblique, acute, and obtuse triangles, as well as angle measurements and trigonometric ratios.
