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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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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = 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
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = 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
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called π (pi). Since the diameter is twice the radius, the circumference of a circle is
The area of a circle is
where r is the radius of the circle.
An arc is part of the circumference of a circle. It is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, 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, like a slice of pizza.
The area of a circle is πr2, and there are 360 degrees of rotation in a circle. Therefore, a sector with central angle θ is 360∘θ of a full circle, and has area
One radian is the interior angle of an arc which has a length equivalent to the radius r of the circle. Since the circumference of a circle is given by 2πr, then, there are 2π total radians in a circle (the equivalent of 360°).
Since the perimeter of a full circle is 2πr, the angle θ corresponding to a full circle (360°) is
So
Some key angles in radians and degrees:
An arc is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, 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:
The area of a circle is πr2, and there are 2π radians in a circle. Therefore, a sector with central angle θ is 2πθ of a full circle, and has area
so
The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.
Take a look at the graph below and notice the following relationships always hold:
The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:
The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.
Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:
Notice that both sinx and cosx have a domain of x∈R and a range of (−1,1).
A sinusoidal function is a generalization of sin and cos to the form
or
The tan function is defined by tanx=cosxsinx.
The domain is thus x=22k+1π (there are vertical asymptotes at those x′s), and the range is all real numbers R.
The function has roots at x=0,±π,±2π… (ie x=kπ where k∈Z)
When we have a trig equation where the argument to the trig function is of the form ax+b, we need to find the domain of ax+b using the domain of x. For example, if 0≤x<2π and we have sin(2x+2π)=1, then
therefore
For any value of θ:
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
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called π (pi). Since the diameter is twice the radius, the circumference of a circle is
The area of a circle is
where r is the radius of the circle.
An arc is part of the circumference of a circle. It is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, 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, like a slice of pizza.
The area of a circle is πr2, and there are 360 degrees of rotation in a circle. Therefore, a sector with central angle θ is 360∘θ of a full circle, and has area
One radian is the interior angle of an arc which has a length equivalent to the radius r of the circle. Since the circumference of a circle is given by 2πr, then, there are 2π total radians in a circle (the equivalent of 360°).
Since the perimeter of a full circle is 2πr, the angle θ corresponding to a full circle (360°) is
So
Some key angles in radians and degrees:
An arc is defined by the radius r of the circle and the angle θ that the arc "sweeps out" over the circle's perimeter.
Since the arc length is a fraction of the overall circumference determined by the value of the angle θ, 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:
The area of a circle is πr2, and there are 2π radians in a circle. Therefore, a sector with central angle θ is 2πθ of a full circle, and has area
so
The key idea with triangles who's hypotenuse lie on the unit circle (that form an angle of θ with the x-axis) is that cosθ represents length of the base, and sinθ represents the height.
Take a look at the graph below and notice the following relationships always hold:
The following table shows the values of sinθ and cosθ for the so called critical angles θ. These are angles that give "nice" values for sin and cos:
The unit circle can be divided into quadrants based on the sign of cosθ and sinθ. These correspond to the 4 quadrants produced by the intersection of the x and y axes. The quadrants are denoted Q1, Q2, Q3 and Q4.
Since a full circle is 2π radians, adding 2π to any angle θ gives the same point on the unit circle. In fact, adding any integer multiple of 2π gives the same point:
Notice that both sinx and cosx have a domain of x∈R and a range of (−1,1).
A sinusoidal function is a generalization of sin and cos to the form
or
The tan function is defined by tanx=cosxsinx.
The domain is thus x=22k+1π (there are vertical asymptotes at those x′s), and the range is all real numbers R.
The function has roots at x=0,±π,±2π… (ie x=kπ where k∈Z)
When we have a trig equation where the argument to the trig function is of the form ax+b, we need to find the domain of ax+b using the domain of x. For example, if 0≤x<2π and we have sin(2x+2π)=1, then
therefore
For any value of θ: