Questions and Answers
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 ratio of the opposite side to the hypotenuse in a right-angled triangle?
What is the period of the sine wave?
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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?
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What type of triangle has one angle that is 90 degrees?
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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.
