©Carolyn C. Wheater, 2000 1

Basis of TrigonometryBasis of TrigonometryBasis of TrigonometryBasis of Trigonometry

Trigonometry, or "triangle measurement," developed as a means to calculate the lengths of sides of right triangles.

It is based upon similar triangle relationships.

©Carolyn C. Wheater, 2000 2

Right Triangle TrigonometryRight Triangle TrigonometryRight Triangle TrigonometryRight Triangle Trigonometry

You can quickly prove that the two right triangles with an acute angle of 25°are similar

All right triangles containing an angle of 25° are similar

25

25

You could think of this as the family of 25° right

triangles. Every triangle in the family is similar.

We could imagine such a family of triangles for any

acute angle.

You could think of this as the family of 25° right

triangles. Every triangle in the family is similar.

We could imagine such a family of triangles for any

acute angle.

©Carolyn C. Wheater, 2000 3

Right Triangle TrigonometryRight Triangle TrigonometryRight Triangle TrigonometryRight Triangle Trigonometry

In any right triangle in the family, the ratio of the side opposite the acute angle to the hypotenuse will always be the same, and the ratios of other pairs of sides will remain constant.

©Carolyn C. Wheater, 2000 4

The Three Main RatiosThe Three Main RatiosThe Three Main RatiosThe Three Main Ratios

If the three sides of the right angle are labeled as the hypotenuse, the side opposite a particular

acute angle, A, and the side adjacent to the acute

angle A,

six different ratios are possible.

A

hypotenuse

adjacent

oppo

site

©Carolyn C. Wheater, 2000 5

The Three Main RatiosThe Three Main RatiosThe Three Main RatiosThe Three Main Ratios

sin( )A opposite

hypotenuse

cos( )A adjacent

hypotenuse

tan( )A opposite

adjacent

SOH

CAH

TOA

A

c

b

a

©Carolyn C. Wheater, 2000 6

Solving Right TrianglesSolving Right TrianglesSolving Right TrianglesSolving Right Triangles

With these ratios, it is possible to solve for any unknown side of the right triangle, if

another side and an acute angle are known, or to find the angle if two sides are known.

Once upon a time, students had to rely on tables to look up these values. Now the sine, cosine, and tangent of an angle can be found on your calculator.

Once upon a time, students had to rely on tables to look up these values. Now the sine, cosine, and tangent of an angle can be found on your calculator.

©Carolyn C. Wheater, 2000 7

Trig TablesTrig TablesTrig TablesTrig Tables

©Carolyn C. Wheater, 2000 8

Sample ProblemSample ProblemSample ProblemSample Problem

In right triangle ABC, hypotenuse is 6 cm long, and A measures 32. Find the length of the shorter leg. Make a sketch If one angle is 32, the other is 58 The shorter leg is opposite the smaller angle, so

you need to find the side opposite the 32 angle.

6

32

58

©Carolyn C. Wheater, 2000 9

Choosing the RatioChoosing the RatioChoosing the RatioChoosing the Ratio

... Find the length of the shorter leg. You need a ratio that talks about

opposite and hypotenuse Can use sine (sin) or cosecant

(csc), but since your calculator has a key for sin, sine is more convenient.

6

32

58

©Carolyn C. Wheater, 2000 10

Solving the TriangleSolving the TriangleSolving the TriangleSolving the Triangle

sin( )326

x

From your calculator, you can find that sin(32) 0.53, so

0 536

. x

x 3 2.

6

32

58