Trig Special

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SECTION 4.1 Special Right Triangles and Trigonometric Ratios

MATH 1330 Precalculus 345

Chapter 4Trigonometric Functions

Section 4.1: Special Right Triangles andTrigonometric Ratios

Special Right Triangles Trigonometric Ratios

Special Right Triangles

Right Triangles:


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CHAPTER 4 Trigonometric Functions

University of Houston Department of Mathematics346

45o-45o-90o Triangles:

Theorem for 45o-45

o-90

oTriangles:

Example:

Solution:


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30o-60o-90o Triangles:

Theorem for 30o-60

o-90

oTriangles:

Example:


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Solution:

Additional Example 1:

Solution:

Part (a):

Part (b):


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Solution:


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Solution:

Part (a):

Part (b):


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Solution:


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Solution:


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Trigonometric Ratios

The Three Basic Trigonometric Ratios:

Example:


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The Three Reciprocal Trigonometric Ratios:


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Example:

Solution:


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Solution:

Part (a):

Part (b):


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Part (c):



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Solution:


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Solution:


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Solution:


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Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios


Answer the following.

1. If two sides of a triangle are congruent, then the__________ opposite those sides are also

congruent.

2. If two angles of a triangle are congruent, then the__________ opposite those angles are also

congruent.

3. In any triangle, the sum of the measures of itsangles is _____ degrees.

4. In an isosceles right triangle, each acute anglemeasures _____ degrees.

5. Fill in each missing blank with one of thefollowing: smallest, largest

In any triangle, the longest side is opposite the

__________ angle, and the shortest side is

opposite the __________ angle.

6. Fill in each missing blank with one of thefollowing: 30o, 60o, 90o

In a 30o-60

o-90

otriangle, the hypotenuse is

opposite the _____ angle, the shorter leg is

opposite the _____ angle, and the longer leg is

opposite the _____ angle.

For each of the following,

(a) Use the theorem for 45

o

-45

o

-90

o

triangles tofindx.

(b) Use the Pythagorean Theorem to verify the

result obtained in part (a).

7.

8.

9.

10.

11.

12.

13.

14.

15.

45o

5x

45o

8

x

45o

x3 2

45o

x

8

45o

x7

9

x

12x

45o

x

4 2

x

8 2


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16.

17.

18.

The following examples help to illustrate the theorem

regarding 30o-60

o-90

otriangles.

19. What is the measure of each angle of anequilateral triangle?

20. An altitude is drawn to the base of the equilateraltriangle drawn below. Find the measures ofx andy.

21. In the figure below, an altitude is drawn to thebase of an equilateral triangle.

(a) Find a and b.(b) Justify the answer obtained in part (a).(c) Use the Pythagorean Theorem to find c, the

length of the altitude.

22. In the figure below, an altitude is drawn to thebase of an equilateral triangle.

(a) Find a and b.(b) Justify the answer obtained in part (a).(c) Use the Pythagorean Theorem to find c, the

length of the altitude. (Write c in simplestradical form.)

For each of the following, Use the theorem for 30o-60

o-

90o

triangles to findx andy.

23.

24.

25.

26.

yo

xo

30o30o

a b

10

c

60o60o

30o30o

a b

4

c

60o60o

30ox

y

7

30o

22

y

x

30o x

8

y

60o

x

y

6 3

45o

2 3

x

45o

x

2 3

x

5 2


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27.

28.

29.

30.

31.

32.

Answer the following. Write answers in simplest form.

33.

(a) Use the Pythagorean Theorem to find BC.

(b) Find the following: sin _____A sin _____B

cos _____A cos _____B

tan _____A

tan _____B

34.

(a) Use the Pythagorean Theorem to find DE.(b) Find the following:

sin _____D sin _____F

cos _____D cos _____F

tan _____D tan _____F

35. Suppose that is an acute angle of a righttriangle and

5sin

7 . Find cos and

tan .

36. Suppose that is an acute angle of a righttriangle and 4 2tan

7 . Find sin and

cos .

37. The reciprocal of the sine function is the_______________ function.

38. The reciprocal of the cosine function is the_______________ function.

39. The reciprocal of the tangent function is the_______________ function.

40. The reciprocal of the cosecant function is the_______________ function.

41. The reciprocal of the secant function is the_______________ function.

42. The reciprocal of the cotangent function is the_______________ function.

60ox

y

5

60oy

15 3

x

30ox

6

y

x

30o

8

y

60ox

y

4 2

60oy5 3

x

24

25D

FE

1517

A

BC


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43.

(a) Use the Pythagorean Theorem to findx.(b) Find the six trigonometric functions of .(c) Find the six trigonometric functions of .

44.


45.


46.



2 10cot

3 . Find the six

trigonometric functions of .


5sec

2 . Find the six

trigonometric functions of .

49.

(a) Use the theorems for special right trianglesto find the missing side lengths in the

triangles above.

(b) Using the triangles above, find thefollowing:

sin 45 _____ csc 45 _____

cos 45 _____ sec 45 _____

tan 45 _____ cot 45 _____

(c) Using the triangles above, find thefollowing:

sin 30 _____ csc 30 _____

cos 30 _____ sec 30 _____

tan 30 _____ cot 30 _____

(d) Using the triangles above, find thefollowing:

sin 60 _____ csc 60 _____

cos 60 _____ sec 60 _____

tan 60 _____ cot 60 _____

6

x

5

8

x4

8

x

6

x

7

4

45o

3

60o

30o

2


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50.

(a) Use the theorems for special right trianglesto find the missing side lengths in the

triangles above.

(b) Using the triangles above, find thefollowing:

sin 45 _____ csc 45 _____

cos 45 _____ sec 45 _____

tan 45 _____ cot 45 _____

(c) Using the triangles above, find thefollowing:

sin 30 _____ csc 30 _____

cos 30 _____ sec 30 _____

tan 30 _____ cot 30 _____

(d) Using the triangles above, find thefollowing:

sin 60 _____ csc 60 _____

cos 60 _____ sec 60 _____

tan 60 _____ cot 60 _____

51. Compare the answers to parts (b), (c), and (d) inthe previous two examples. What do you notice?

45o

8

60o

30o12

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