Questions and Answers
What type of triangle has one angle that is 90 degrees?
What is the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle?
What type of angle is greater than 180 degrees but less than 360 degrees?
What is the period of the tangent function?
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What type of triangle has all sides of equal length?
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What is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse in a right triangle?
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What is the range of the sine function?
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What is the relationship between the tangent, sine, and cosine functions?
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What is the relationship between two angles that add up to 90°?
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What is the definition of the cosine of an angle in a right-angled triangle?
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What is the range of the tangent function?
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What is the Pythagorean theorem used for?
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What is the reciprocal identity of the cosine function?
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What type of triangle has no right angles?
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What is the definition of the sine of an angle in a right-angled triangle?
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What is the graph of the sine function?
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Study Notes
Triangles
- A triangle is a polygon with three sides and three angles.
- Triangles can be classified into different types based on their angles and side lengths:
- Equilateral triangle: all sides are equal
- Isosceles triangle: two sides are equal
- Scalene triangle: all sides are unequal
- Right triangle: one angle is 90 degrees (a "right angle")
- Oblique triangle: no angle is 90 degrees
Angles
- An angle is formed by two rays sharing a common endpoint (vertex).
- Angles can be measured in degrees, radians, or revolutions.
- Angles can be classified into different types:
- Acute angle: less than 90 degrees
- Right angle: exactly 90 degrees
- Obtuse angle: greater than 90 degrees but less than 180 degrees
- Straight angle: exactly 180 degrees
- Reflex angle: greater than 180 degrees but less than 360 degrees
Sine (sin)
- The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse (the side opposite the right angle).
- sin(A) = opposite side / hypotenuse
- The sine function has a range of [-1, 1] and a period of 360 degrees.
Cosine (cos)
- The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- cos(A) = adjacent side / hypotenuse
- The cosine function has a range of [-1, 1] and a period of 360 degrees.
Tangent (tan)
- The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
- tan(A) = opposite side / adjacent side
- The tangent function has a range of (-∞, ∞) and a period of 180 degrees.
- tan(A) = sin(A) / cos(A)
Triangles
- A triangle has three sides and three angles.
- Triangles can be classified into five types based on their angles and side lengths:
- Equilateral triangle: all sides are equal.
- Isosceles triangle: two sides are equal.
- Scalene triangle: all sides are unequal.
- Right triangle: one angle is 90 degrees.
- Oblique triangle: no angle is 90 degrees.
Angles
- An angle is formed by two rays sharing a common endpoint (vertex).
- Angles can be measured in degrees, radians, or revolutions.
- Angles can be classified into five types:
- Acute angle: less than 90 degrees.
- Right angle: exactly 90 degrees.
- Obtuse angle: greater than 90 degrees but less than 180 degrees.
- Straight angle: exactly 180 degrees.
- Reflex angle: greater than 180 degrees but less than 360 degrees.
Trigonometric Ratios
Sine (sin)
- The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- sin(A) = opposite side / hypotenuse.
- The sine function has a range of [-1, 1] and a period of 360 degrees.
Cosine (cos)
- The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- cos(A) = adjacent side / hypotenuse.
- The cosine function has a range of [-1, 1] and a period of 360 degrees.
Tangent (tan)
- The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
- tan(A) = opposite side / adjacent side.
- The tangent function has a range of (-∞, ∞) and a period of 180 degrees.
- tan(A) = sin(A) / cos(A).
Angles
- Angles can be measured in degrees, radians, or gradians
- Acute angles are less than 90°
- Right angles are exactly 90°
- Obtuse angles are greater than 90° but less than 180°
- Straight angles are exactly 180°
- Reflex angles are greater than 180° but less than 360°
- Complementary angles add up to 90°
- Supplementary angles add up to 180°
- Vertical angles are formed by two intersecting lines
Trigonometric Ratios
- Sine (sin) is the ratio of the opposite side to the hypotenuse in a right-angled triangle
- Cosine (cos) is the ratio of the adjacent side to the hypotenuse in a right-angled triangle
- Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle
- Sine, cosine, and tangent are the three basic trigonometric ratios
- Cosecant (csc) is the reciprocal of sine
- Secant (sec) is the reciprocal of cosine
- Cotangent (cot) is the reciprocal of tangent
- Pythagorean identities relate sine, cosine, and tangent
Triangles
- Right-angled triangles have one right angle (90°)
- Pythagorean theorem: a² + b² = c² for right-angled triangles
- Oblique triangles have no right angles
- Law of sines: a / sin(A) = b / sin(B) = c / sin(C) for oblique triangles
- Law of cosines: c² = a² + b² - 2ab * cos(C) for oblique triangles
Trigonometric Identities
- Sum and difference formulas relate trigonometric ratios of angles
- Double and half-angle formulas relate trigonometric ratios of angles
- Examples of sum and difference formulas:
- 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))
- Examples of double and half-angle formulas:
- sin(2A) = 2 * sin(A) * cos(A)
- cos(2A) = cos²(A) - sin²(A)
- tan(2A) = 2 * tan(A) / (1 - tan²(A))
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Description
Learn about the different types of triangles, including equilateral, isosceles, scalene, right, and oblique triangles, and understand the concept of angles in geometry.
