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Here are a few recommended readings before getting started with this lesson.
Many angles can be drawn on a coordinate plane by placing the vertex in the origin and choosing two rays. By this method, it is possible to have different angles with the same measure. In the applet, move the rays around to form an angle with a measure of $60_{∘}.$ Then, try getting the same measure, but with the rays in entirely different positions than the first time.
Different angles with a shared vertex at the origin can be formed such that they all have the same measure. This can be somewhat vague, however. To remove this ambiguity, one of the sides of the angle is fixed and an orientation is chosen.
Move the slider to see how the angle measure changes.
Dylan always wakes up late for school and misses the bus. He wants to design a clock with an alarm on it. Dylan brings a rough sketch of the design to his Auntie Wilma, an engineer. Seeing the numbers are not accurately placed, she teaches Dylan how to properly distribute the numbers of the clock.
Knowing this, Dylan erases all the numbers on his design and starts over. Auntie Wilma recommends that he draw a coordinate plane on top of the design so that the origin is at the center of the clock.
Which numeral should Dylan draw at $0_{∘}$ in standard position?An angle of $0_{∘}$ in standard position is on the $x-$axis. Use the fact that the numerals are $30_{∘}$ apart from each other.
Dylan starts by remembering that the initial side of an angle in standard position is on the $x-$axis. Since the measure of the angle is $0_{∘},$ the terminal side of the angle lies on top the initial side.
The numeral at $0_{∘}$ is the one on the right hand side of a clock. By looking at one of Auntie's clocks, Dylan figures out that the numeral is $3.$ He goes ahead and writes it down on his clock design.
According to Auntie Wilma, numerals are $30_{∘}$ apart from each other. Thus, to find the numeral at $60_{∘}$ in standard position, Dylan counts how many $30_{∘}$ intervals are in $60_{∘}.$Once more, Dylan checks one of Auntie's clocks and finds that $1$ is the numeral at $60_{∘}.$ So, he writes it down. So far, his design looks as follows.
Dylan sets out some art supplies to begin painting the clock's housing. He checks another one of Auntie Wilma's clocks as he begins painting. It is $3:00PM.$
Finished and exhausted, Dylan checks the clock again. He is surprised to see that it is now $8:00PM.$ What a long painting session!
Auntie Wilma sees this as an opportunity to teach Dylan about measurements and asks him to find the measure of the angle, in degrees and radians, made by the displacement of the hour hand.
Relation between Radians and Degrees |
$180_{∘}=πrad$ |
$LHS/6=RHS/6$
Calculate quotient
$LHS⋅(-5)=RHS⋅(-5)$
Multiply
$a⋅cb =ca⋅b $
Depending on how the rotation of its terminal side was performed, it is possible to assign different measures to an angle in standard position. These angles, which have the same terminal side, have a special name.
Although Dylan has just measured angles counterclockwise, Auntie Wilma suggests that he should be able to measure angles clockwise. Dylan knows that the angle between the $3$ and $1$ numeral measures $60_{∘},$ his goal is to express that angle in the same direction as the movement of the clock's hands.
Find the negative coterminal angle, in degrees, that is closest to $60_{∘}.$Coterminal angles can be found by adding or subtracting multiples of $360_{∘},$ or $2π$ rad, from the given angle.
Dylan wants to find an angle that is coterminal to a $60_{∘}$ angle that is measured in the same direction as the movement of the clock's hands. This means that he is looking to measure it clockwise. That means the measure will be negative in value.
Coterminal angles can be found by adding or subtracting multiples of $360_{∘}$ to a given angle. Since Dylan is looking for the negative coterminal angle closest to $60_{∘},$ the coterminal angle can be found by subtracting $360_{∘}$ from $60_{∘}.$
Relation between Radians and Degrees |
$180_{∘}=πrad$ |
$LHS/3=RHS/3$
Calculate quotient
$LHS⋅(-5)=RHS⋅(-5)$
One of the goals of this lesson is to associate trigonometric ratios to any angle. However, these are defined for acute angles, so a way around must be chosen. Quadrantal and reference angles are used for this purpose.
Degrees | Radians | |
---|---|---|
Quadrant I | $θ_{′}=θ$ | $θ_{′}=θ$ |
Quadrant II | $θ_{′}=180_{∘}−θ$ | $θ_{′}=π−θ$ |
Quadrant III | $θ_{′}=θ−180_{∘}$ | $θ_{′}=θ−π$ |
Quadrant IV | $θ_{′}=360_{∘}−θ$ | $θ_{′}=2π−θ$ |
If $θ$ is greater than $360_{∘}$ or less than $0_{∘},$ then finding its coterminal angle with a positive measure between $0_{∘}$ and $360_{∘}$ will be helpful to find the reference angle.
Angle | Coterminal Angle | Reference Angle |
---|---|---|
$-130$ | $-130+360=230$ | $230−180=50$ |
$38π $ | $38π −2π=32π $ | $π−32π =3π $ |
Dylan is still working on pinpointing the clock numerals. To help himself, he draws a coordinate plane on top of his design, just as Auntie Wilma suggested.
Auntie Wilma says to Dylan that now he should draw the numerals corresponding to the quadrantal angles. What numerals should Dylan draw?Identify which numerals are on the axes of the coordinate plane. Find the numerals that are next to the ones on the $x-$axis.
The terminal side of a quadrantal angle lies on the $x-$ or $y-$axis. Therefore, the numerals that Dylan should draw are those that lie on top of the coordinate axes, except for the numeral $3,$ which he drew earlier.
Once more, Dylan looks at one of Auntie's clock and sees that the numerals corresponding to quadrantal angles are $3,$ $6,$ $9,$ and $12.$ However, he already drew $3.$ Hence, he proceeds to draw these three new numerals in his design.
To find the numerals that form reference angles of $30_{∘},$ Dylan first reviews the definition of reference angle.
Reference Angle |
For an angle $θ$ that is not quadrantal, the acute angle $θ_{′}$ formed by the terminal side of $θ$ and the $x-$axis is called a reference angle. |
Note that this is the angle between the $x-$axis and the corresponding numeral. This means that it does not have to be an angle in standard position; it can also be either clockwise or counterclockwise.
By looking at the clock on the living room wall of Auntie's house, Dylan finds that the numerals which form a $30_{∘}$ reference angle are $2,$ $4,$ $8,$ and $10.$ He goes ahead and adds those numerals to the design.
An angle in standard position is given in the following applet.
By placing a right triangle with one vertex at the origin and one of their legs along the $x-$axis, the coordinates of $the$ $vertex$ $not$ $on$ $the$ $axis$ can be found using the legs of the triangle.
If said vertex is in the first quadrant, then its coordinates are the lengths of the triangle's legs. Considering that lengths cannot be negative, the sign of the corresponding coordinate changes accordingly if the vertex is not on the first quadrant.Since the hypotenuse of these triangles is $1,$ the marked points all lie on a circle with radius $1$ whose center is at the origin.
This circle receives a special name and is related to trigonometric ratios, particularly to sine and cosine.
The unit circle is a circle with radius $1$ and whose center lies on the origin of a coordinate plane.
A right triangle can be associated to a point on the circle. This can be done by choosing its hypotenuse equal to the radius of the circle. The legs of the triangle are parallel to the axes.
The coordinates of a point on the unit circle can be related to trigonometric ratios by selecting $θ$ as the angle that is at the origin.
The length of the hypotenuse is equal to one because it is the radius of the unit circle, so the point on the unit circle can be used to generalize sine and cosine functions.Quadrantal Angle | Sine | Cosine |
---|---|---|
$0_{∘}$ | $0$ | $1$ |
$90_{∘}$ | $1$ | $0$ |
$180_{∘}$ | $0$ | $-1$ |
$270_{∘}$ | $-1$ | $0$ |
Dylan is ready to finish the clock. He places it, without the housing, in his works place, then sets out a huge sheet of pink graph paper over the already placed numerals. He placed each numeral $1$ foot away from the clock's center.
Dylan wants to place the rest of the numerals, but he has forgotten a protractor to measure the angles! Auntie Wilma points out that his clock can be seen as a unit circle, therefore, the rest of the numerals can be placed by finding their coordinates.
Find the coordinates where Dylan should place the numeral $11.$ Round each coordinate to two decimal places.Numeral | Angle | Reference Angle |
---|---|---|
$11$ | $120_{∘}$ | $60_{∘}$ |
$5$ | $300_{∘}$ | $60_{∘}$ |
Numeral | Angle | Reference Angle | $x-$coordinate | $y-$coordinate |
---|---|---|---|---|
$11$ | $120_{∘}$ | $60_{∘}$ | $cos120_{∘}=-cos60_{∘}$ | $sin120_{∘}=sin60_{∘}$ |
$5$ | $300_{∘}$ | $60_{∘}$ | $cos300_{∘}=cos60_{∘}$ | $sin300_{∘}=-sin60_{∘}$ |
Numeral | $x-$coordinate | $y-$coordinate |
---|---|---|
$11$ | $-0.5$ | $≈0.87$ |
$5$ | $0.5$ | $≈-0.87$ |
Now that Dylan knows the coordinates of the numerals, he proceeds and draws them. Right before drawing the last numeral, $7,$ Auntie Wilma asks him to let her do it, and Dylan happily agrees.
Dylan feels so proud to have made it this far in the clock design. Thanks to viewing his clock as a unit circle, he could draw the last three numerals. Dylan is supremely confident that they are correctly placed. He puts on the finishing touches, removes some of the tics for a more modern look, and sets up this huge and funny looking alarm clock by his bed.
The tangent and the secant are undefined for $0_{∘},$ $180_{∘},$ or any of their coterminal angles. |
Likewise, the cotangent and cosecant are undefined when $y=0,$ that is, for the quadrantal angles on the $y-$axis, as well as for any of their coterminal angles.
The cotangent and the cosecant are undefined for $90_{∘},$ $270_{∘},$ or any of their coterminal angles. |