Trigonometric Functions - Precalculus Chapter 4 - Math … · SECTION 4.1 Special Right Triangles...

SECTION 4.1 Special Right Triangles and Trigonometric Ratios

MATH 1330 Precalculus 345

Chapter 4 Trigonometric Functions

Section 4.1: Special Right Triangles and

Trigonometric Ratios

Special Right Triangles


Special Right Triangles

Right Triangles:

CHAPTER 4 Trigonometric Functions

University of Houston Department of Mathematics 346

45o-45o-90o Triangles:

Theorem for 45o-45

o-90

o Triangles:

Example:

Solution:



30o-60o-90o Triangles:

Theorem for 30o-60

o-90

o Triangles:

Example:



Solution:

Additional Example 1:

Solution:

Part (a):

Part (b):




Solution:




Solution:

Part (a):

Part (b):




Solution:




The Three Basic Trigonometric Ratios:

Example:



Solution:



The Three Reciprocal Trigonometric Ratios:



Example:

Solution:




Solution:

Part (a):

Part (b):



Part (c):




Solution:

Exercise Set 4.1: Special Right Triangles and Trigonometric Ratios


45o

8

x

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 its

angles is _____ degrees.

4. In an isosceles right triangle, each acute angle

measures _____ degrees.

5. Fill in each missing blank with one of the

following: 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 the

following: 30o, 60o, 90o

In a 30o-60o-90o triangle, 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 45o-45

o-90

o triangles to

find x.

(b) Use the Pythagorean Theorem to verify the

result obtained in part (a).

7.

8.

9.

10.

11.

12.

13.

14.

15.

45o

5

x

45o

8

x

45o

x

4 2

45o

x

3 2

45o

x

8

45o

x

7

9

x

12 x

45o

x

4 2

45o

x

4 2

x

8 2



16.

17.

18.

The following examples help to illustrate the theorem

regarding 30o-60

o-90

o triangles.

19. What is the measure of each angle of an

equilateral triangle?

20. An altitude is drawn to the base of the equilateral

triangle drawn below. Find the measures of x and

y.

21. In the figure below, an altitude is drawn to the

base 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 the

base 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 simplest

radical form.)

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

o-

90o triangles to find x and y.

23.

24.

25.

26.

yo

xo

30o

30o

a b

10

c

60o 60o

30o

30o

a b

4

c

60o 60o

30o

x

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



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 right

triangle and 5

sin7

. Find cos and

tan .


triangle and 4 2

tan7

. 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.

60o

x

y

5

60o

y

15 3

x

30o

x

6

y

x

30o

8

y

60o

x

y

4 2

60o

y 5 3

x

24

25 D

F E

15 17

A

B C



43.

(a) Use the Pythagorean Theorem to find x.

(b) Find the six trigonometric functions of .

(c) Find the six trigonometric functions of .

44.




45.




46.





triangle and 2 10

cot3

. Find the six

trigonometric functions of .


triangle and 5

sec2

. Find the six

trigonometric functions of .

49.

(a) Use the theorems for special right triangles

to find the missing side lengths in the

triangles above.

(b) Using the triangles above, find the

following:

sin 45 _____ csc 45 _____

cos 45 _____ sec 45 _____

tan 45 _____ cot 45 _____

(c) Using the triangles above, find the

following:

sin 30 _____ csc 30 _____

cos 30 _____ sec 30 _____

tan 30 _____ cot 30 _____

(d) Using the triangles above, find the

following:

sin 60 _____ csc 60 _____

cos 60 _____ sec 60 _____

tan 60 _____ cot 60 _____

6

x

5

8

x 4

8

x

6

x

7

4

45o

3

60o

30o

2



50.

(a) Use the theorems for special right triangles

to find the missing side lengths in the

triangles above.

(b) Using the triangles above, find the

following:

sin 45 _____ csc 45 _____

cos 45 _____ sec 45 _____

tan 45 _____ cot 45 _____

(c) Using the triangles above, find the

following:

sin 30 _____ csc 30 _____

cos 30 _____ sec 30 _____

tan 30 _____ cot 30 _____

(d) Using the triangles above, find the

following:

sin 60 _____ csc 60 _____

cos 60 _____ sec 60 _____

tan 60 _____ cot 60 _____

51. Compare the answers to parts (b), (c), and (d) in

the previous two examples. What do you notice?

45o

8

60o

30o

12

Trigonometric Functions - Precalculus Chapter 4 - Math … · SECTION 4.1 Special Right Triangles...

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