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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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π = included in formula booklet β’ π« = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
The area of a triangle is given by
where βbβ is the base and βhβ is the height.
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In a right angled triangle with sides βa, βbβ and hypotenuse (longest side) βc, Pythagoras' Theorem states
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In a right angled triangle with an angle βΞΈ<90Β°, the trigonometric ratios βsin, βcosβ and βtanβ are defined by
where opposite and adjacent refer to the side lengths of the sides opposite and adjacent to βΞΈ, while hypotenuse is the length of the longest side.
If we know the value of βsinΞΈ, βcosΞΈβ or βtanΞΈβ in a right angled triangle, we can find βΞΈβ using an inverse trigonometric function on a calculator. These functions are βsinβ1, βcosβ1β and βtanβ1β and satisfy
whenever βΞΈ<90Β°, which is always true in a right angled triangle.
The trigonometric ratios βsin, βcosβ and βtanβ are actually functions that relate an angle βΞΈβ to a ratio of sides. The values of βsin, βcosβ and βtanβ for specific angles can be found on the calculator. For example
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When we know two side lengths and the angle between the two sides, we can find the area even if we don't directly know the height by using the fact that
Thus
The previously found formula for area
applies to any pair of sides and the angle between them. Since the area is the same no matter which sides we use:
Multiplying everything by β2β and dividing by βabc:
Flipping the numerator and denominator gives the form that appears in the formula booklet:
The sine rule is primarily used when we know two angles and a side. When we know two sides and an angle, the version with angles in the numerator is easier to work with.
The cosine rule is a generalization of Pythagoras' theorem for non-right-angled triangles. It states that
The cosine rule is primarily used when we
know two sides and the angle between them, and want to find the third side,
know all three sides and want to find an angle.
The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called βΟβ (pi). Since the diameter is twice the radius, the circumference of a circle is
The area of a circle is
where βrβ is the radius of the circle.
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is βΟr2, and there are β360β degrees of rotation in a circle. Therefore, a sector with central angle βΞΈβ is β360°θββ of a full circle, and has area
One radian is the interior angle of an arc which has a length equivalent to the radius βrβ of the circle. Since the circumference of a circle is given by β2Οr, then, there are β2Οβ total radians in a circle (the equivalent of β360Β°β).
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Since the perimeter of a full circle is β2Οr, the angle βΞΈβ corresponding to a full circle (β360Β°β) is
So
Some key angles in radians and degrees:
An arc is defined by the radius βrβ of the circle and the angle βΞΈβ that the arc "sweeps out" over the circle's perimeter.
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Since the arc length is a fraction of the overall circumference determined by the value of the angle βΞΈ, the arc length is calculated as
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is βΟr2, and there are β2Οβ radians in a circle. Therefore, a sector with central angle βΞΈβ is β2ΟΞΈββ of a full circle, and has area
so
Angles of elevation and depression describe how far upward or downward you look relative to a horizontal line when observing an object.
The angle of elevation is the angle formed by looking upward from the horizontal line to an object above your line of sight.
The angle of depression is the angle formed by looking downward from the horizontal line to an object below your line of sight.
These angles are always measured relative to a horizontal line, never vertical. Because the lines of sight form alternate interior angles with horizontal lines, the angle of elevation from one viewpoint equals the angle of depression from the other viewpoint.
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A true bearing describes a direction measured clockwise from the north direction (β0Β°β) around a full circle up to β360Β°. Bearings are always given using three digits (e.g. β045Β°,120Β°,270Β°β) to avoid confusion.
A bearing of β000ββ points directly north.
β090ββ points east.
β180ββ points south.
β270ββ points west.
When working with true bearings, clearly draw a compass rose to visualize directions and measure angles clockwise from the north line.
One very useful fact about bearings is that returning in the direction something came from means adding or subtracting β180Β°β from the original bearing - since it is doing a U-turn.
Whether you add or subtract depends on whether the original bearing is smaller or bigger than β180Β°, since the resulting bearing must be between β0β and β360Β°.
A sphere is a perfectly round, three-dimensional geometric shape where every point on its surface is exactly the same distance (the radius) from a single central point. It's the three-dimensional analog of a circle. For example, a ball or globe is spherical in shape. The surface area βAβ and volume βVβ of a sphere are given by:
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A cylinder is a three-dimensional geometric shape formed by two identical circular bases connected by a curved lateral surface. The segment connecting the centers of the circular bases is called the axis, which is perpendicular to each base in a right cylinder (the type usually studied).
The volume βVβ of a cylinder with radius βrβ is given by:
The curved surface of a cylinder (excluding the circular ends) is given by:
If we include the circular ends, each with area βΟr2, we get
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A prism is a three-dimensional solid shape consisting of two parallel, congruent faces called bases, connected by rectangular lateral faces. Prisms are named according to the shape of their basesβfor example, triangular prism, rectangular prism, or hexagonal prism.
The volume βVβ of a prism is calculated by multiplying the area βAβ of its base by its height βh:
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A right circular cone is a three-dimensional geometric shape whose apex (vertex) lies directly above the center of its circular base.
Key Parts:
Circular Base: Flat circle with radius βr.
Apex (Vertex): The point directly above the center of the base.
Height (βhβ): Perpendicular distance from apex to base center.
Slant Height (βlβ): Distance along the cone's surface from apex to edge of base.
Formulas:
Volume:
Surface Area of curved surface:
Slant Height Relationship:
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A right pyramid is a three-dimensional shape with a polygonal base and triangular lateral faces, in which the apex (vertex) is located directly above the center (centroid) of the base.
Key Parts:
Polygonal base: a flat polygon (triangle, square, pentagon, etc.)
Apex (vertex): the point positioned vertically above the base's centroid
Height (βhβ): perpendicular distance from apex to base centroid
Slant height (βlβ): distance along a lateral face from the apex perpendicular to an edge of the base
The volume of a right pyramid is given by
where βAβ is the area of the base.
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