Lesson 3 More “Sine” Language Solidify Understanding

Ready

1.

The circle has been divided into equal portions. Explain the meaning of each number in the equation .

a circle is sliced into 12 even sections. the angle of each section at the center point is labeled radian.
Untitled the measure of the central angle θ,or the angle of the intercepted arcthe whole circle divided bythe number of equal portionsthe number of degrees ina full revolution of a circle

In Geometry, we learned how to find the arc length of a portion of a circle when we are given the measure of the intercepted arc in degrees and the radius of the circle. We can find the fraction of the circle using , where is the measure of the intercepted arc. Then multiply the fraction of the circle times the circumference or .

Use the information in each drawing to find the length of the intercepted arc on the circle.

2.

a circle is graphed on a coordinate plane with a center point of a and a radius of 7 feet. There is a right angle drawn within the circle. A0

3.

a circle is graphed on a coordinate plane with a center point of a and a radius of 12 centimeters. There is an angle of 135 degrees drawn within the circle. A0

4.

a circle is graphed on a coordinate plane with a center point of a and a radius of 9 centimeters. There is an angle drawn in 3 of the 4 quadrants. A0

5.

a circle is graphed on a coordinate plane with a center point of a and a radius of 8 inches. There is an angle of 343 degrees drawn within the circle. A0

Set

A right triangle has been drawn from a point on a circle centered at . The angle of rotation, , has been drawn between the initial ray and the terminal ray.

  • Write the value of sine as defined in today’s lesson. (You will need to find the radius.)

  • Use right angle trig to solve for angle inside the triangle.

  • Use the measure of to find (the angle of rotation).

6.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from W to the point (4,3) creating a right angle on the x axis and an undetermined angle in the center. wwwAF

a.

b.

c.

7.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from W to the point (-4, 4 time the square root of 3) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. AF

a.

b.

c.

8.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from W to the point (- square root of 2 over 2, - square root of 2 over 2) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. AF

a.

b.

c.

9.

a circle is graphed on a coordinate plane with a center point of W. A ray is drawn from W to the point (- 6 time the square root of 3, - 6) creating a right angle on the x axis and an undetermined angle from the hypotenuse to the x axis. AF

a.

b.

c.

10.

Describe the angles of rotation that make the -values of the points be positive and the angles of rotation that make the -values be negative.

11.

What do you notice about the -values and the value of sine in the previous graphs?

12.

In the graph shown, the radius of the circle is unit. The intersections of the circle and the axes are labeled. Based on your observation in #11, what do you think the value of sine might be for the following values of ?

a circle graphed on a coordinate plane with points at (-1,0), (0,-1), (1,0), and (0,1). (1, 0)(1, 0)(1, 0)(0, 1)(0, 1)(0, 1)(-1, 0)(-1, 0)(-1, 0)(0, -1)(0, -1)(0, -1)AF

a.

:

b.

:

c.

:

d.

:

Go

Make a sketch of the following problems, then solve.

13.

A kite is aloft at the end of a string. The string makes an angle of with the ground.

a.

Sketch.

b.

How far above the ground is the kite? (Round your answer to the nearest whole number.)

14.

A ladder leans against a building. The top of the ladder reaches a point on the building that is above the ground. The foot of the ladder is from the building.

a.

Sketch.

b.

Find to the nearest degree the measure of the angle that the ladder makes with the level ground.

15.

The shadow of a flagpole is long when the angle of elevation of the sun is .

a.

Sketch.

b.

Find the height of the flagpole.

16.

The angle of depression from the top of a building to a car parked in the parking lot is . The building is high.

a.

Sketch.

b.

How far from the top of the building is the car on the ground?