Download Lesson 6-1 Introduction to Trigonometric Functions

Math 2 Honors
Lesson 6-1: Introduction to Trigonometric Functions
Name ________________________
Learning Goals:
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

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I can use the Pythagorean Theorem to solve for an unknown side length of a right triangle.
I can use the characteristics of similar figures to justify the trigonometric ratios.
I can define the following trigonometric ratios for acute angles in a right triangle, sine, cosine,
and tangent.
I can calculate sine and cosine ratios for acute angles in a right triangle when given two side
lengths.
1. Study the diagram below of a radio transmitter tower with two support wires attached to it at A and E
and to the ground at B and D. First, focus on the triangle formed by the tower, the ground, and the
shorter support wire.
a. If CD = 50 feet and CE = 100 feet, is the size and shape of right DCE completely determined?
Why or why not? (“Is the triangle determined?” means can more than one triangle with those
dimensions be created. Thinking back to the last lesson, are all triangles with the given
information congruent?)
b. How long is support wire DE ?
c. If you were asked to attach a support wire, 125 feet long, from point D to the tower, how far up
the tower would you have to climb?
d. What theorem among sides of a right triangle have you used in Parts b and c?
2. Now focus on the right triangle formed by the tower, BC , and the longer support wire AB.
a. If BC = 75 feet and mABC  66, is the size and shape of right ABC completely determined?
Why or why not?
b. Explain why the size and shape of a right triangle are completely determined when the measures
of two sides are known or the measure of one acute angle and the length of one side are known.
c. It should be possible to use the information given in Part a to find AC, the distance above the
ground at which support wire AB is attached to the tower. Will the Pythagorean Theorem help?
Explain.
To solve problems like Problem 2 Part c, you need to find a connection between angle measures and
segment lengths in a right triangle. In this case, it is helpful to think of an angle as being formed by
rotating a ray about its endpoint from an initial position to a terminal position.
The point about which the ray is rotated is the vertex of the angle. The
initial position of the ray is the initial side of the angle. The terminal
position of the ray is the terminal side of the angle. To indicate the
direction of rotation from initial side to terminal side, it is customary to say
the angle has positive measure if the rotation is counterclockwise and has
negative measure or the rotation is clockwise.
3. The diagram at the right shows four points on the terminal side
of an angle in standard position in a coordinate plane. The
vertex of the angle is at the origin and its initial side coincides
with the positive x-axis.
a. For each point (x, y) shown on the terminal side, find the ratios
i.
U (8, 6)
iii. W (2,1.5)
b. How do the ratios
ii.
y
.
x
V (12, 9)
iv. Z (4, 3)
y
compare in each case? Why does that make sense?
x
c. For each point (x, y) in Part a, suppose r is the distance from the origin to the point. How do you
y
x
think the ratios
would compare in each case? The ratios ? Check your conjectures.
x
y
i. r 
x

r
y

r
ii. r 
x

r
y

r
iii. r 
x

r
y

r
iv. r 
x

r
y

r
What do you notice about
x
y
and ?
r
r
If P ( x, y ) is a point (not on the origin) on the terminal side of an angle in standard position and
y y
x
r  x 2  y 2 , then the ratios , , and
do not depend on the choice of P; they depend only on the
r
x r
measure of POQ. That is, these ratios are functions of the measure  of the angle. These functions
are called trigonometric functions and are given special names as indicated below.
tangent of   tan  
sine of   sin  
y
( x  0)
x
y
r
cosine of   cos  
x
r
4. The terminal side of an angle in standard position with measure  contains the given point. In each
case, draw the angle on a coordinate plane. The find cos  , sin  , and tan  .
a. P (12, 5)
cos  
sin  
tan  
b. P ( 6, 4)
cos  
sin  
tan  
c. For any angle with measure  , is it possible for sin   1? For cos   1? For tan   1?
Answer with yes or no, then explain why.
sin  
cos  
tan  
5. The diagram below shows a portion of a circle with radius 10 cm, drawn on a 2-mm grid. Angles
are marked off in 10 intervals, so mAOP1  10, mAOP2  20, mAOP3  30, and so on. You
can use this diagram to calculate approximate values of cos  , sin  , and for angles with measure 
between 0 and 90
a. Verify the entries in the table below for the case of AOP4 .
b. Use the diagram to find the missing entries in the table.
6. In Problem 5, you were able to find approximate values of the trigonometric functions of 20, 40,
60, and 80. In the case of angles with measure 0 and 90, you were able to determine exact
values (with the exception of tan 90 ). You can use geometric reasoning to find exact values of
tangent, sine, and cosine of 45.
a. Draw an angle of 45 in standard position.
b. What is an equation for the line that makes an angle of 45
with the positive x-axis?
c. What do you know about the coordinates of any point on the terminal side of this angle?
(a , )
d. Choose a point (x, y) on the terminal side of the angle and then find tan 45, sin 45, and
cos 45. Give exact values, not decimal approximations.
tan 45o 
sin 45o 
cos 45o 
7. Now return to Problem 2 about the radio transmitter tower. Suppose BC  75 feet and
mABC  66.
To determine AC, the distance above the ground at which support wire AB is attached to the tower:
a. Draw ABC in standard position as in the diagram above on the right. Explain why the
coordinates of points A and C are labeled as shown.
b. Write an expression for tan 66 in terms of the given information.
tan 66o 
c. Use the diagram in Problem 5 to calculate an approximate value of tan 66.
d. Using your results from Parts b and c, find the approximate height AC.
e. How long is support wire AB ?
tan 66o  _______
AC =________
AB =_______
f. Show how you could use a trigonometric function to find the approximate length of AB without
first finding the height AC. Compare you answer to that obtained in Part e.
8. The values of the trigonometric functions you found by using the grid in Problem 5 were rough
approximations of the function values. Several centuries ago, mathematicians spent years
calculating values of the trigonometric functions by hand to several decimal placed so they could be
used in surveying and astronomy. Today, more advanced mathematical methods are used. You will
study those methods in a later course. A portion of a table of trigonometric function values with
four-digit accuracy is shown below.
a. Compare the values of sine, cosine, and tangent of 45 that you found in Problem 6 with the
values in the table above.
b. Compare the values of tan 66 and cos 66 that you used in Problem 7 with the values in the
table above.
c. As the measure of an angle increases from 45 to 75,
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?
d. Why do the patterns of change in Part c make sense in terms of the diagram in Problem 5?
e. Use the table to determine each of these trigonometric function values.
i. sin 58o =
ii. cos 48o =
iii. tan 72o =
f. Given the function values below, use the table to determine the measure of the angle  whose
terminal side is in the first quadrant.
i. sin   0.8910
ii. cos   0.5878
iii. tan   1.1106