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Algebra 2 9-12 / Week 09 / Friday
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Week 09 · Trigonometry on the Unit Circle

Friday

Around and around
// Every angle has a point
⏱ about 20 min

Friday: Around and Around

Comet swings the string a full turn and keeps going, stopping 30° past the start. "390 degrees. What now?"

"Same point as 30°," Wren says. "You went once around and then 30 more. The circle does not remember the lap."

"So sin 390° is sin 30°," Comet says. "And if I swing backwards?" She turns the string clockwise 60°.

"Negative 60°," Wren says. "Same point as 300°. Cosine positive, sine negative."

Nova draws a spiral of angles landing on the same mark. "Would you like a hint? Add or subtract 360 until you land between 0 and 360."

"Then every real number is an angle," Comet says. "Every one gets a point, and every point gives a sine and a cosine."

"That is why the wave next week never ends," Wren says.

Beyond one turn, and backwards

An angle can be any real number of radians or degrees. Counterclockwise is positive, clockwise is negative.

Angles that differ by a full turn, 360° or 2π, land on the same point. They have the same sine and cosine.

To find the point, add or subtract full turns until the angle lies between 0° and 360°. Then use the reference angle.

AngleSame point asPoint (cos θ, sin θ)
390°30°(√3/2, 1/2)
-60°300°(1/2, -√3/2)
450°90°(0, 1)
750°30°(√3/2, 1/2)
-120°240°(-1/2, -√3/2)

Where the crew used the circle this week

QuestionToolAnswer
How long is a 60° arc on the 1 m circle?radians as arc lengthπ/3 ≈ 1.05 m
What is cos 150°?reference angle and quadrant-√3/2
sin θ = 3/5 in quadrant 2; cos θ?the Pythagorean identity-4/5
What is sin 390°?subtract a full turn1/2

Which answer makes sense?

Comet measured a mark at x = 0.71 and y = -0.71. Wren asked which angle. Two answers fit the size: 45° or 315°.

The negative y picks quadrant 4, so 315°. The numbers alone are not enough; the signs decide.

MIXED SET
  • Read the question.
  • Tap your answer.
Which angle between 0° and 360° ends in the same place as 390°?
Which angle between 0° and 360° ends in the same place as -60°?
A unit circle with a 30° angle marked from the positive x-axis; the point on the circle has coordinates (cos 30°, sin 30°).What is sin 390°?
A unit circle with a 300° angle marked from the positive x-axis; the point on the circle has coordinates (cos 300°, sin 300°).What is cos -60°?
THE WEEK IN FOUR QUESTIONS
  • Read the question.
  • Tap your answer.
What is 270° in radians?
What is 5π/6 radians in degrees?
A unit circle with a 210° angle marked from the positive x-axis; the point on the circle has coordinates (cos 210°, sin 210°).What is sin 210°?
sin θ = 3/5 and θ ends in quadrant 4. What is cos θ?
Which angle between 0° and 360° lands on the same point as 750°? Type the number.
How many degrees is π/3 radians? Type the number.
A 180° arc on the 1 m chalk circle is how many meters? Use 3.14 for π. Type the number.
StatementTrue or false?
390° and 30° land on the same point of the unit circle.?
390 - 360 = 30?
A negative angle means turning clockwise.?
sin(-60°) is positive.?
Every real number can be an angle with a point on the unit circle.?
WHY THIS EXERCISEThe circle turns a right-triangle idea into a function of every real number. That is the big move of the week.
Draw the unit circle and mark the points for 30°, 390° and -330° in one color. Then mark -60° and 300° in another.

A full lap and more. Tomorrow is Garden Day, and the family chalks a circle of its own.

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