Download Lesson 6-1 Practice - Mayfield City Schools

Math 2 Honors
Lesson 6-1 Practice
Name_________________________
1. The terminal side of an angle in standard position with measure  contains the point P(4, 7).
a. Draw a sketch of the angle.
b. Find sin  , cos  , and tan  .
c. Use the table in Lesson 5-1 Problem 8 to estimate  to the nearest degree.
2. More people these days are exercising regularly. Exercise scientists measure the amount of work
done by people in various forms of exercise so they can learn more about its effect. One popular
form of exercise is walking on a treadmill.
a. What features of a treadmill do you think would increase or decrease the amount of work done
by the walker?
b. One index that exercise scientists use is the percent grade of the treadmill. Percent grade is
computed as 100 multiplied by the tangent of the measure of the angle of elevation  of the
treadmill. Suppose  is in standard position. Compute the percent grade of a two-meter (axleto-axle) treadmill with a vertical rise of 0.25 meters. Of 0.33 meters.
c. How do you think the percent grade is related to the amount of work a person does on a
treadmill?
3. Steep hills on highways are the scourge of long-distance bikers. To measure the percent grade of a
section of highway, surveyors use transits to estimate the average angle of elevation (or inclination)
over a measured distance of highway. Then the percent grade is computed in the same way as
described in Problem 2 Part b for a treadmill.
a. If you ride down a straight 3-mile section of highway that has an 8% grade, how far do you drop
vertically if the horizontal distance is 2.987 miles?
b. If the angle of inclination of a 2-mile section of straight highway is about 4, what is the percent
grade?
4. Suppose the terminal side of an angle in standard position with measure  contains the indicated
point. Find sin  , cos  , and tan  . Then find the measure of the angle to the nearest degree.
a. P(3, 4)
b.
c. P (0,  10)
d. P( 5, 5)
P (5, 12)
5. The diagram below extends the diagram from Lesson 5-1 to show several other angles, AOPn , in
standard position in a coordinate plane. Angles are marked off in 10 intervals.
a. Use the diagram to calculate approximate values of the following:
i. cos120
ii.
sin120
iii.
tan120
iv. cos160
v.
sin160
vi.
tan160
b. What is cos180 ? sin180 ? tan180 ?
c. How could you use symmetry of the semicircle and you completed copy of the table from
Lesson 5-1 Problem 5 to determine each of the trigonometric function values in Part a?
d. Refer to the diagram below. As the measure of AOPn increases from 90 to 180,
i.
how does the sine of the angle change?
ii.
how does the cosine of the angle change?
iii.
how does the tangent of the angle change?
6. A portion of a circle with radius 1 and center at the origin is shown below. Point P1 is the image of
point A under a 45 counterclockwise rotation about the origin. Point P2 is the image of point A
under a counterclockwise rotation of  degrees about the origin.
a. What is the length of OP1 ? Of OP2 ? Explain your reasoning.
b. Why does a  cos 45 ? Why does b  sin 45 ?
c. What is the x-coordinate c of point P2 written as a trigonometric function of  ?
What is the y-coordinate d?
7. In Lesson 5-1, you used geometric reasoning to determine exact values of sin 45, cos 45, and
tan 45. You also determined exact values of these functions for angles of measure 0 and 90
(with the exception of tan 90 ). Use equilateral triangle ABC to help you determine the exact
trigonometric function values below. AM is a median of the triangle. (The side length 2 is chosen
for ease of computation.)
a. cos 60 =
b. sin 60 =
c. tan 60 =
d. cos 30 =
e. sin 30 =
f. tan 30 =
8. A line through the origin forms an acute angle with the positive x-axis of degree measure  called
the angle of inclination of the line. Express the slope of the line as a trigonometric function of  .
What is the equation of the line?
9. Suppose ABC is a right triangle with C a right angle.
a. If mA  50, what is mB ?
If mB  10, what is mA ?
b. If you know the measure of one acute angle of a right triangle, explain why you can always
determine the measure of the other acute angle.
c. Two angles whose measures sum to 90 are called complementary angles. In what sense do
they complement each other?
d. The term “cosine” suggests that the cosine of an acute angle is the sine of the complement of that
angle. Does cos 50  sin 40 ? Does cos80  sin10 ?
e. Suppose D and E are acute angles of a right triangle.
i.
Use a diagram to explain why cos D  sin E.
ii.
Why can the equations in part i also be written as cos D  sin(90  D ) ? Why can it
also be written as sin E  cos(90  E ) ?