Angles and their measure

Angular Measurement and Circular Sectors

A section of a circle is known as a sector. One side of the sector is the radius. The portion of the cirumference that is included in the sector is known as an arc.

Angular Degree

A circle has 360 degrees. Each degree can have 60 minutes (designated as ' ) and each minute can have 60 seconds (designated as " ). Since we can not convert minutes or seconds into radians directly we need to convert the minutes and seconds into a decimal number. Here is the formula:

Convert $X^{\circ }$ Y' Z" into degrees. $X+\left(Y\times {\frac {1}{60}}\right)+\left(Z\times {\frac {1}{3600}}\right)=X+{\frac {Y}{60}}+{\frac {Z}{3600}}$ For a practical example, convert $23^{\circ }$ 18' 38" into degrees. $23+{\frac {18}{60}}+{\frac {38}{3600}}=23.3106$

Classification of Angles by Degree Measure

Acute Angle

• an angle is said to be acute if it measures between 0 and 90 degrees, exclusive.

Right Angle

• an angle is said to be right if it measures 90 degrees.
• notice the small box placed in the corner of a right angle, unless the box is present it is not assumed the angle is 90 degrees.
• all right angles are congruent

Obtuse Angle

• an angle is said to be obtuse if it measures between 90 and 180 degrees, exclusive.

Angular Radian

A radian is the angle subtended at the centre of a circle by an arc of its circumference equal in length to the radius of the circle. Since we know that that the formula for the circumference of a circle is $C=2\pi r$ we can determine that there are $2\pi$ radians in a circle. We abbreviate radians as rad.

Conversion Between Degrees and Radians

Mathematics requires us to use radians for most angular measurements. Therefore we need to know how to convert from degrees to radians Since we know that there are $360^{\circ }$ degrees or $2\pi$ radians in a circle. We can determine these equations: $1^{\circ }={\frac {\pi }{180}}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\ radian = \frac {180}{\pi }}

So we can write these general formulae.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^\circ = \frac {X \times \pi }{180}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\ radian \left(s \right ) = \frac {X \times 180}{\pi }}

Here is an example convert Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 20^\circ} into radians

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 20^\circ = \frac {20 \times \pi }{180} = \frac{1}{9} \pi\ rad}

Convert Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{9} \pi} into degrees.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{9} \pi\ radian \left(s \right ) = \frac {\frac{1}{9} \pi \times 180}{\pi } = 20^\circ }

Arc Length

In most cases it is very difficult to measure the length of an arc with a ruler. Therefore we need to use a formula in order to determine the length of the arc. The formula that we use is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Arc\ Length = ( \theta\ in\ radians)(radius)\,} in symbols this is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s= \theta r\,} . Note: θ need to be in radians

Here is an example, determine the length of the arc created by a sector with a 6 cm radius and an angle of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 53^\circ} .

The first thing we need to do is convert θ from degrees to radians.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 53^\circ = \frac {53 \pi}{180}rad}

Now we can calcuate the length of the arc.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = \frac {53 \pi}{180} \times 6 \approx 5.55cm}

Area of a Sector

The area of a sector can be found using this formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Area = \frac{1}{2}r^2 \theta} Note: θ need to be in radians.

Calculate the area of a sector with a 3 cm radius and an angle of 2π.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Area = \frac{1}{2}3^2 \times 2 \pi \approx 28.3cm^2}

The unit circle

The center of the unit circle is the origin Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} , point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=(x_A,y_A)} . Creating line segments starting from point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis will create two line segments. From this, we learn that, in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} , point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=(\cos\theta,\sin\theta)} and other properties.

A circle with the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2=1} is called a unit circle. Imagine a circle with center at the origin of a Cartesian coordinate system.

Rotating a ray from the direction of the positive half of the x-axis by an angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} (counterclockwise for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta > 0,} and clockwise for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta < 0} ) yields intersection points of this ray with the unit circle: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{A} = (x_\mathrm{A},y_\mathrm{A})} , and, by extending the ray to a line if necessary, with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{“}x=1\text{”}:\;\mathrm{B} = (x_\mathrm{B},y_\mathrm{B}),} and with the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{“}y=1\text{”}:\;\mathrm{C} = (x_\mathrm{C},y_\mathrm{C}).} The tangent line to the unit circle in point A, which is orthogonal to this ray, intersects the y- and x-axis at points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{D} = (0,y_\mathrm{D})} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{E} = (x_\mathrm{E},0)} . The coordinate values of these points give all the existing values of the trigonometric functions for arbitrary real values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} in the following manner:

Trigonometric functions and the unit circle

Function	Right-triangle definition	Unit circle definition
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin\theta}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\textrm{opposite}}{\textrm{hypotenuse}}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_A}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\theta}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\textrm{adjacent}}{\textrm{hypotenuse}}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_A}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan\theta}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\textrm{opposite}}{\textrm{adjacent}}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_B=\frac{y_A}{x_A}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc\theta}	${\frac {\textrm {hypotenuse}}{\textrm {opposite}}}$	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_D=\frac{1}{y_A}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec\theta}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\textrm{hypotenuse}}{\textrm{adjacent}}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_E=\frac{1}{x_A}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot\theta}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\textrm{adjacent}}{\textrm{opposite}}}	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_C=\frac{x_A}{y_A}}

The importance of the unit circle is that it expands the definitions of trigonometric functions. Before, those values are constrained within a right triangle. Now, the functions can be used on any radian as their input.

Resources

• Angles and their measure. Written notes created by Professor Esparza, UTSA.
• Trigonometry, Wikibooks: Calculus
• Angles, Wikibooks: Geometry
• Circles and Angles, Wikibooks: A-Level Mathematics