Angles, the Foundation For Trig (5.1)

Vocabulary,

Location,

Rotation

POD

If a spinner starts in the third quadrant, and spins in a negative direction for 1 ¼ revolutions, what quadrant does it land in?

Vocabulary (some review and some new)

We can think of an angle as two line segments with a common end point, or two rays with a shared end point. We can also think of them as a rotation of rays. What does that rotation mean?

What symbol do we use to indicate an angle?

Coterminal angles start and end in the same locations– they have the same initial and terminal sides, no matter the direction or number of rotations.

What would a straight angle be?

AO

B

Initial side

Terminal side

Vocabulary

We use a number of lower case Greek letters to represent angles:

α (alpha), β (beta), γ (gamma), θ (theta), φ (phi), and so on.

You will see all of these, especially as we do trig identities, but if it gets too complicated, you can also substitute any of these symbols with x. (Then give the answer with the original symbol.)

Location

An angle is in standard position on the coordinate plane when its vertex is on the origin and its initial side is along the positive x-axis.

An angle is a quadrantal angle if it is in standard position and its terminal side lies on an axis.

Rotation

What limit, if any, is there to rotation?

Which direction is positive and which is negative?

Rotation– acute and obtuse

Rotations are measured several ways. We will use primarily degrees and radians. If you do not see a degree symbol (°), the measure is given in radians. Radian measure often, but not always, includes π.

What constitutes a full revolution in either unit?

Give the range for acute angles using degree and radian measure.

Give the range for obtuse angles using degree and radian measure.

Rotation—complementary and supplementary

Rotations are measured several ways. We will use primarily degrees and radians. If you do not see a degree symbol (°), the measure is given in radians. Radian measure often, but not always, includes π.

Give the rule for complementary angles α and β in degree and radian measure.

Give the rule for supplementary angles γ and θ in degree and radian measure.

Rotation—conversion

Be careful when using the calculator! What do you need to check first?

It’s straightforward to convert between degree and radian measure.Conversion units:

π/180° (degree to radian) 180°/π (radian to degree)

Give an equivalent measure in radians for 72°.

Give an equivalent measure in degrees for 1.25π radians.

Rotation—multiples

Give an equivalent measure for a single rotation, coterminal angle to an angle of 63°. For two rotations. How large could the equivalent measure be? Could it be negative? How could you express the measure in general?

Do the same for an angle of π/4.

Measures

Degree measures are broken down into smaller units: minutes and seconds.

Notation: degrees °, minutes’ , and seconds”.

There are 60 minutes in a degree, and 60 seconds in a minute: 1°= 60’ = 3600”.

Your calculator computes decimal degrees into these units.

Change 171.8873° into degrees, minutes and seconds.

Enter the degree amount, then 2nd-ANGLE-4-ENTER.

Arcs and sectors

An arc is the section of a circle “cut out” from a given central angle.

A sector is the interior of the circle “cut out” from a given central angle.

The measure of each is based on a ratio of the central angle to a full rotation:

arc length = θ/360°(circumference)

sector area = θ/360°(area)

Arcs and sectors

An arc is the section of a circle “cut out” from a given central angle.

A sector is the interior of the circle “cut out” from a given central angle.

In radian measure, we can shorten the formulas:

arc length = (θ/2π)(2πr) = θr

sector area = (θ/2π)(πr2) = (1/2)θr2

Finally… radian exploration

We talk about radian measure a great deal. But what does “one radian” actually mean?

By tables, get a circle, 2 or 3 Wiki Stix, a piece of paper, and a colored pencil. Follow the next directions.

Finally… radian exploration

Find a way to determine the center of your circle. Think back to your geometry days. Mark the center with a dry erase marker.

Using that center, measure the radius of your circle with a Wiki Stik, folding it to the right length.

How many of those shorter radius lengths equal the distance around the circle? How does this match up with the formula for the circumference of a circle?

Finally… radian exploration

Trace your circle onto the piece of paper. Again, estimate the center of the circle.

Starting at one point on the circle, run the radius-length Wiki Stik along the circle you traced and mark where it starts and stops.

Now, make an angle from the start of that string to the center of the circle and to the end of the string. Color in that sector.

What do you notice about your angle with others’?

Finally… radian exploration

The measure of each angle corresponds to the arc length of one radius. That matching angle is said to be one radian in measure. Get it?

An angle of one radian spins out an arc of one radius in length. One radian angle means one radius arc. No matter the size of the circle.