Learning Objective:

To be able to describe the sides of right-angled triangle for use in trigonometry.

•Setting up ratios•Trig in the Calculator

Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.

Angle

A

A

The sides of a right -angled triangle are given special names:The hypotenuse, the opposite and the adjacent.The hypotenuse is the longest side and is always opposite the right angle.The opposite and adjacent sides refer to another angle (given to us), other than the 90o.

There are three formulae involved in trigonometry:

sin A=

cos A=

tan A =

S O H C A H T O A

Finding the ratios

The simplest form of question is finding the decimal value of the ratio of a given angle.

Find:

1) sin 32 =sin 32 =

2) cos 23 =

3) tan 78 =

4) tan 27 =

5) sin 68 =

Using ratios to find angles

It can also be used in reverse, finding an angle from a ratio.To do this we use the sin-1, cos-1 and tan-1 function keys. (hitting the 2nd key first)

Example:1. sin x = 0.1115 find angle x.

x = sin-1 (0.1115)x = 6.4o

2. cos x = 0.8988 find angle x

x = cos-1 (0.8988)x = 26o

sin-1 0.1115 =shift sin( )

cos-1 0.8988 =shift cos( )

Ex. 1: Finding Trig Ratios• Compare the

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

15

817

A

B

C

7.5

48.5

A

B

C

Ex. 1: Finding Trig Ratios

15

817

A

B

C

7.5

48.5

A

B

C

Large Small

sin A =

opposite hypotenuse

cosA =

adjacent hypotenuse

tanA =

opposite adjacent

817

≈ 0.4706

1517

≈ 0.8824

815

≈ 0.5333

48.5

≈ 0.4706

7.58.5

≈ 0.8824

47.5

≈ 0.5333

Trig ratios are often expressed as decimal approximations.

Ex. 2: Finding Trig Ratios

S

sin S =

opposite hypotenuse

cosS =

adjacent hypotenuse

tanS =

opposite adjacent

513

≈ 0.3846

1213

≈ 0.9231

512

≈ 0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle.

R

sin S =

opposite hypotenuse

cosS =

adjacent hypotenuse

tanS =

opposite adjacent

1213

≈ 0.9231

513

≈ 0.3846

125

≈ 2.4

adjacent

opposite12

13 hypotenuse5

R

T S

Examples of Trig Ratios

Sin P

Cos P

1220

16Q

P

Tan P Tan Q

Cos Q

Sin Q1620

1220

1612

1220

1620

1216

Opposite

Similar Triangles and Trig Ratios

ABC QPR

35

4A

B12

20

16Q

P

RC

Tan Q

Cos Q

Sin Q1220

1620

1216

Tan A

Cos A

Sin A35

45

34