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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.
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:
Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.
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)
Since sinθ represents the y-coordinate of a point on the unit circle, solving the equation
is equivalent to drawing the line y=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.
This helps visualize all the possible solutions. For
the solutions are
Since cosθ represents the x-coordinate of a point on the unit circle, solving the equation
is equivalent to drawing the line x=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.
This helps visualize all the possible solutions. For
the solutions are
Since tanθ represents the angle between the line y=xtanθ and the x-axis, solving
is equivalent to drawing the line y=a, and measuring the minor and major angles it forms with the x-axis:
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
Trigonometric functions can also show up in pseudo-quadratics - a quadratic where the variable being squared is not x but a trig function.
On exams, these equations often require using the Pythagorean identity sin2θ+cos2θ=1.
For any value of θ:
The double angle identity for sin states that
The double angle identity for cosine states that
These three different forms come from leveraging sin2θ+cos2θ=1.
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.
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:
Once the values of sin and cos are memorized for critical angles in the first quadrant, you can use the rules of symmetry and periodicity to determine the values of critical angles in arbitrary quadrants.
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)
Since sinθ represents the y-coordinate of a point on the unit circle, solving the equation
is equivalent to drawing the line y=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.
This helps visualize all the possible solutions. For
the solutions are
Since cosθ represents the x-coordinate of a point on the unit circle, solving the equation
is equivalent to drawing the line x=a, finding the intersections with the unit circle, and determining the angle of these intersections relative to the positive x-axis.
This helps visualize all the possible solutions. For
the solutions are
Since tanθ represents the angle between the line y=xtanθ and the x-axis, solving
is equivalent to drawing the line y=a, and measuring the minor and major angles it forms with the x-axis:
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
Trigonometric functions can also show up in pseudo-quadratics - a quadratic where the variable being squared is not x but a trig function.
On exams, these equations often require using the Pythagorean identity sin2θ+cos2θ=1.
For any value of θ:
The double angle identity for sin states that
The double angle identity for cosine states that
These three different forms come from leveraging sin2θ+cos2θ=1.