Warm Up for Section 1.2 Simplify: (1). (2). (3). There are 10 boys and 12 girls in a Math 2 class. Write the ratio of the number of girls to the number.

Warm Up for Section 1.2

Simplify: (1). (2).

(3). There are 10 boys and 12 girls in a Math 2 class. Write the ratio of the number of girls to the number of boys as a fraction in simplest form.

(4). In a 30o-60o-90o triangle, the hypotenuse measures 10 inches. What is the length of the long leg?

1252

72

Answers for Warm Up Section 1.2

Simplify: (1). (2).

(3). There are 10 boys and 12 girls in a Math 2 class. Write the ratio of the number of girls to the number of boys as a fraction in simplest form.

(4). In a 30o-60o-90o triangle, the hypotenuse measures 10 inches. What is the length of the long leg?

1252

72310 14

5

6

35

Work for Answers to WU, Section 1.2

(1). (2).

(3). girls (4). boys

310

325

345125

2

142

2

72

2

72

2

2

14

5

6

10

12

30o

60o

10

x

x23x 5

102

x

x

35

Trigonometric RatiosSection 1.2

Standard: MM2G2

Essential Question: What is the definition of the sine, cosine, and tangent ratios?

Investigation 1:

(1). How do you find the missing side length?

108

x

222 108 x

100642 x

362 x

6x

Use the Pythagorean Theorem to findthe missing side length!

(2). Does that work for this triangle? Why not?

31o

5

To use the Pythagorean Theorem, we must know 2 of the 3 side lengths. We only know one side length, so this theorem cannot be used.

We need something besides the Pythagorean Theorem to find missing measures for right triangles when we don’t know two side lengths. That something is Trigonometry!

Trigonometry is a branch of mathematics that deals with the relationships between the sides of angles of triangles and the calculations based on these relationships.

A trigonometric ratio describes a relationship between the sides and angles of a right triangle.

(3). It is important to identify the correct side when creating a trig ratio. Let’s investigate ABC below and its sides: a, b, and c.

With your partner, identify each of the following:

hypotenuse: _______

side opposite angle A: _______

side adjacent to angle A: _______

side opposite angle B: _______

side adjacent to angle B: _______

A

BC

b

a

c

c

a

b

b

a

Recall: Similar triangles are triangles that have the same shape but not necessarily the same size. (4). In similar triangles, corresponding angles are congruent and corresponding sides are proportional. Consider the four similar triangles pictured below.

How do you know the triangles are similar?

Ans: Because corresponding angles are congruent

For the 31o angle in each triangle, write the ratio of :adj

opp

3 5

9 15

6 10

33 55= = =

Now, find the missing side, x, by setting up a proportion and solving? Then, how could you find y?

31o

8

x y

x

8

5

3

403 x

3

40x

Once we know x, we can use it and 8 in the Pythagorean Theorem to find y.

Early mathematicians found these ratios of similar triangles very useful and they named them the Trigonometric Ratios. They made tables of these calculations. Today, we can obtain these values with a calculator!

The ratios we will use are listed below – memorize them!

tangent of θ = Length of leg opposite θLength of leg adjacent to θ

Length of leg opposite θLength of hypotenuse

Length of leg adjacent to θLength of hypotenuse

For any acute angle in a right triangle, we denote the measure of the angle by θ and define three ratios related to θ as follows:

sine of θ =

cosine of θ =

In the figure below, the terms “opposite,” “adjacent,” and “hypotenuse” are used as shorthand for the lengths of these sides. Using this shorthand, we can give abbreviated versions of the above definitions:

tan =

sin =

cos =

oppositeadjacent

oppositehypotenuse

adjacenthypotenuse SOH-CAH-TOA

The Indian princess will help you remember !!

Check for Understanding: Write each trigonometric ratio for the right triangle pictured below:

sin A = ____=___ tan A = ____=___ cos B = ____=___

cos A = ___=____ sin B = ___=____ tan B = ___=___

15

12

9

A

BC

1215

12 9

1215

915

915

912

45

45

43

34

35

35

(5). sin P = (6). sin S = (7). sin L =

(8). cos P = (9). cos T = (10). cos M =

(12). tan P = (13). tan S = (14). tan L =

13

T

12

5

SU

108

P

QR 6

1715

L

NM 8

610

810

68

513

513

512

817

817

815

Investigation 2: Use your calculator to estimate each trig ratio to the nearest thousandth (three digits after the decimal). Follow the steps below to estimate each value.

Step 1: Set you calculator to degree mode.

Step 2: Press the desired ratio (tan, sin, cos), type the angle measure in degrees, press ENTER.

Step 3: Round your answer to three digits after the decimal.

(14). cos(47o) ____________

(15). sin(63o) _____________

(16). tan(12o) ______________

0.682

0.891

0.213

Investiagion 3: Write and solve an equation using a trig ratio for 48o using 10 and x. Use your calculator to estimate the value of the trig ratio to three digits after the decimal. Round your answer to three digits as well.

10

48

x

tan (48o) = x10

10 (tan 48o) = x

11.106 x

Check for Understanding: For each diagram, write a trig ratio for the given acute angle using x and the given side. Solve for x. Round answers to the nearest thousandth.

(17).

x20

50°

cos (50o) = 20 x

x (cos 50o) = 20

x =

x 31.114

20cos 50o

20°

x

12

(18).

sin (20o) =

12 (sin 20o) = x

x 12

4.104 x

(19).

12

x25°

tan(25o) =

x =

12 x

x ≈ 25.7

)25tan(

12

Formula Sheet:

Trigonometric Ratios: SOH-CAH-TOA

sin θ =

cos θ =

tan θ =

hyp

opp

hyp

adj

adj

opp

θ

Opp

Adj

Hyp