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### Trig Ratios with Complementary Angles and Similar Triangles

Warm Up for Section 1.4

Find the missing edge lengths:

(1). (2).

45o

60o

45o

30o

Find the value of x to the nearest tenth:

(3). (4).

46o

x

x

63o

Answers for Warm Up Section 1.4

Find the missing edge lengths:

(1). (2).

45o

60o

45o

30o

Find the value of x to the nearest tenth:

(3). (4).

46o

x

x

63o

Standard:MM2G2ab

Section 1.4

Essential Question: What is the relationship

between the trig ratios of complementary angles

and similar triangles?

Review: With your partner answer each question

reviewing the information we have learned in

Sections 1.1-1.3:

(1). Label each side of the triangle at

right as the hypotenuse, adjacent

side to ,or sideoppositeangle .

(2). Using the terms “opposite,” “adjacent,” and

“hypotenuse,” write each trig ratio for the

triangle. (Abbreviate please!)

(a). cos = ______ (b). tan = ______

(c). sin = ______

hypotenuse

opposite

adjacent

Opp

Adj

Adj

Hyp

Opp

Hyp

Investigation 1: Use Triangles #1, #2, and #3 below

to complete the chart at the bottom of the page:

In each cell of the chart you will write the indicated

trig ratio for the specified angle. DO NOT simplify

each fraction.

Triangle #1 Triangle #2 Triangle #3

(3). What is the sum of the measures a and b

in each right triangle? ____o

(4). So, these angles are referred to as

______________ angles.

(5). If a = 70o then b = _____o.

(6). Now refer to the values in the chart.

Are any of values exactly the same?

____

90

complementary

20

Yes!

(7). Record the trig functions that have equivalent

values for each triangle. The first example has

been done for you for Triangle #1.

Triangle #1

__________ = __________

__________ = __________

Triangle #2 Triangle #3

__________ = __________ _______ = ______

__________ = __________ ________ = _______

sin a cos b

sin b cos a

sin a cos b

sin a cos b

sin b cos a

sin b cos a

Summary: For each pair of complementary angles in a

right triangle, the sine of one angle is the cosine of its

_____________.

complement

Check for Understanding:

(8). sin (30o)= cos (____) (9). cos(20o)= sin(____) (10). sin(12o) = cos (____)

In general: sin = cos (_________)

cos = sin (_________)

(11). If angle A is the complement of angle B and

then cos B = ______.

(12). If angle C is the complement of angle D and

cos D = 0.887, then sin C = ________.

70o

60o

78o

90o –

90o –

sin A

0.887

Investigation 2: The multiplicative inverse, or

reciprocal, of any real number a where a 0 is .

So, the reciprocal of 2 is and the reciprocal of

is _____.

Look at the chart again and the values for the

tangent ratio in each triangle. Fill in the blank for

each triangle. The first blank has been done for

you.

Triangle #1 Triangle #2

__________ = __________ __________ = __________

__________ = __________ __________ = __________

Triangle #3

__________ = __________

__________ = __________

tan a

tan a

tan b

tan b

Summary:For each pair of

complementary angles in a right triangle,

the tangent of one angle is the

____________ of the tangent of its

complement.

tan a

reciprocal

tan b

Check for Understanding:

(13). tan (20o) = _______ (14). tan (40o) = ______

(15). tan (32o) = _________

In general, tan = _____________.

(16). If A and B are complementary angles and

tan A = , then tan B = _______.

(17). If C and D are complementary angles and

tan C = 2.2, then tan D = .

Write an identity statement using the complement of the given angle.

(18). sin 52o = (19). cos 29o=

(20). tan 60o = (21). cos 75o =

(22). tan 10o = (23). sin 67o =

cos 38o

sin 61o

sin 15o

cos 23o

Y

15

5

9

4

A

Z

X

12

3

B

C

Investigation 3:

(24). Do ∆ABC and ∆XYZ appear to be similar triangles? _____

YES!

(25). Let’s check the ratio of the corresponding sides:

Y

A

15

9

5

4

Z

X

12

3

B

C

(27).

sin A = sin X =

cos A = cos X =

tan A = tan X =

(28). Are the sine, cosine, and tangent ratios for

corresponding angles of similar triangles the same?

YES!

39

36

R

15

T

S

Check for Understanding: Given:∆RST ∆PYQ

Q

52

20

Y

P

48

Fill in each blank with the corresponding angle:

(29). sin R = sin ___ (30). cos Q = cos ___

(31). tan T = tan ___

P

T

Q

39

36

R

15

T

S

Check for Understanding: Given:∆RST ∆PYQ

Q

52

20

Y

P

48

Write each ratio as a fraction in simplest form:

(32). sin R = _______ and sin P = _______

(33). tan T = _______ and tan Q = _______

(34). If DOG CAT with right angles at O and at

A , then sin D = ______, tan T = ______,

and cos G = ______.

Classwork Answers:

Write each trig expression as an equivalent

expression involving the complement of the angle.

(35). sin(45o) = ________ (36). cos(50o) = _______

(37). tan(22o) = ________ (38). cos(70o) = _______

(39). tan(31o) = _________ (40). sin(49o) = _______

sin C

tan G

cos T

cos (45o)

sin (40o)

1

sin (20o)

tan (68o)

1

cos (41o)

tan (59o)

Angle A is the complement of angle B. Find each of the following.

(41). (42).

(43). (44).

(45).(46). (47).

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