Lesson 9.5 Trigonometric Ratio

Lesson 9.5 Trigonometric Ratio

• Learning Target: I can solve the missing side using trigonometric function.

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

Finding Trigonometric Ratios

The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

The word trigonometry is derived from the ancient Greek language and means measurement of triangles.

TRIGONOMETRIC RATIOS

B

CA

c

b

a

hypotenuse sideoppositeA

side adjacent to A

sin A = = ac

side opposite Ahypotenuse

cos A = = bc

side adjacent Ahypotenuse

tan A = =ab

side opposite Aside adjacent to A

The value of the trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.


Let be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.

ABC


Compare the sine, the cosine, and the tangent ratios for A in each triangle below.

SOLUTION

Their corresponding sides are in proportion, which implies that the trigonometric ratios for A in each triangle are the same.

Large triangle Small triangle

sin A =opposite

hypotenuse

cos A =adjacent

hypotenuse

tan A =oppositeadjacent

0.47068

17 0.47064

8.5

0.88241517 0.8824

7.58.5

0.53338

15 0.53334

7.5

Trigonometric ratios are frequently expressed as decimal approximations.

A

B

C

178

15

A

B

C

8.54

7.5By the SSS Similarity Theorem, the triangles are similar.


Find the sine, the cosine, and the tangent of the indicated angle.

S

R

T S

513

12SOLUTION

The length of the hypotenuse is 13. For S, the length of the opposite side is 5, and the length of the adjacent side is 12.

sin S 0.3846=5

13

cos S 0.9231=1213

tan S 0.4167=5

12

opp.

adj.

hyp.

R

T S

5

12

13

opp.hyp.

=

adj.hyp.

=

opp.adj.

=


Find the sine, the cosine, and the tangent of the indicated angle.

R

R

T S

513

12SOLUTION

The length of the hypotenuse is 13. For R, the length of the opposite side is 12, and the length of the adjacent side is 5.

sin R 0.9231=1213

cos R 0.3846=5

13

tan R = 2.4=125

adj.

opp.

hyp.

opp.hyp.

=

adj.hyp.

=

opp.adj.

=

R

T S

5

12

13

Trigonometric Ratios for 45º

Find the sine, the cosine, and the tangent of 45º.

SOLUTION

Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg.

1

1

45º

hyp.

tan 45º = 1=11

sin 45º 0.7071

cos 45º

opp.hyp.

=

adj.hyp.

= 0.7071

opp.adj.

=

From the 45º-45º-90º Triangle Theorem, it follows that the length of the

hypotenuse is 2 .

= 1 2

= 22

= 1 2

= 22

2

Trigonometric Ratios for 30º

Find the sine, the cosine, and the tangent of 30º.

SOLUTION

1

30º

0.5774

= 0.5=12

2

0.8660

tan 30º

sin 30º

cos 30º

opp.hyp.

=

adj.hyp.

=

opp.adj.

=

To make the calculations simple, you can choose 1 as the length of the shorter leg.

From the 30º-60º-90º Triangle Theorem, it follows that the length of the longer leg is

3 and the length of the hypotenuse is 2.

32

=

= 1 3

33

= 3

Using a Calculator

You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.

Sample keystroke sequencesSample calculator

displayRounded

approximation

3.4874

0.2756

0.96130.96126169574 or 74sin Entersin

74 or 74cos cos Enter

74 or 74tan tan Enter

0.275637355

3.487414444

The sine or cosine of an acute angle is always less than 1.

Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1.

The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse.

The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.


The sine or cosine of an acute angle is always less than 1.

Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1.

TRIGONOMETRIC IDENTITIES

(sin A) 2 + (cos A)

2 = 1

tan A =sin Acos A A

B

C

ca

b


A trigonometric identity is an equation involving trigonometric ratios that is true for all acute angles. The following are two examples of identities:

The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse.

The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.

Using Trigonometric Ratios in Real Life

Suppose you stand and look up at a point in the distance, such as the top of the tree. The angle that your line of sight makes with a line drawn horizontally is called the angle of elevation.

Indirect Measurement

FORESTRY You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of a tree. You measure the angle of elevation from a point on the ground to the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.

tan 59° =oppositeadjacent

45 tan 59° = h

45(1.6643) h

74.9 h

The tree is about 75 feet tall.

Write ratio.

Substitute.

Multiply each side by 45.

Use a calculator or table to find tan 59°.

Simplify.

tan 59° =oppositeadjacenth45

Estimating a Distance

ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg length of 76 feet.

sin 30° =opposite

hypotenuse

d sin 30° = 76

d = 152

A person travels 152 feet on the escalator stairs.

Write ratio for sine of 30°.

Substitute.

Multiply each side by d.

Divide each side by sin 30°.

Simplify.

sin 30° =opposite

hypotenuse76d

d =76

sin 30°

d =760.5

Substitute 0.5 for sin 30°.

30°

76 ftd

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Lesson 9.5 Trigonometric Ratio