Geometry 9.5 Trigonometric Ratios May 5, 2015Geometry 9.5 Trigonometric Ratios w/o Calculator2 Goals I can find the sine, cosine, and tangent of an acute.

Geometry

9.5 Trigonometric Ratios

April 18, 2023 Geometry 9.5 Trigonometric Ratios w/o Calculator 2

Goals

I can find the sine, cosine, and tangent of an acute angle.

I can solve problems using trigonometric ratios.


Terminology

No one, uses the word trigonometry.

It’s


What is trig?

Literally, the measure of triangles. An extremely useful, practical and

powerful math tool. A branch of math that finds its way into

practically everything we do. Usually learned in high school.


What you will learn…

The basic terms and methods of solving right triangles.

How to solve problems using trig.


Trig Ratios

Based on the sides of a right triangle. We will study only three: Sine Cosine Tangent


Right Triangle

Hypotenuse

Leg

Leg

A

From A, this leg is the Adjacent side.

From A, this leg is the Opposite side.


Right Triangle

Hypotenuse

Leg

Leg

A

From A, this leg is the Adjacent side.

From A, this leg is the Opposite side.

B

From B, this leg is the Adjacent side.

From B, this leg is the Opposite side.


Right Triangle

Hypotenuse

Adjacent

Opposite

A


Trig Ratio Definition: Sine

Hypotenuse

Adjacent

Opposite

A

Sine of A = Opposite

Hypotenuse


Trig Ratio Definition: Cosine

Hypotenuse

Adjacent

Opposite

A

Cosine of A = Adjacent

Hypotenuse


Trig Ratio Definition: Tangent

Hypotenuse

Adjacent

Opposite

A

Tangent of A = OppositeAdjacent


Abbreviations

Tangent of A = OppositeAdjacent

Sine of A = Opposite

Hypotenuse

Cosine of A = Adjacent

Hypotenuse

sin A

cos A

tan A


Memory Aid

Sine is Opposite over Hypotenuse. Cosine is Adjacent over Hypotenuse. Tangent is Opposite over Adjacent.

SOHCAHTOA


Writing Ratios SOHCAHTOA

4sin

53

cos54

tan3

B

B

B

3

4

5

A

B 3sin

54

cos53

tan4

A

A

A

?

??

??

?

?

??

??

?


Find these values using your calculator—Make sure you’re in Degree Mode!!!

SIN 15 COS 45 TAN 45 COS 80 SIN 10 TAN 5 COS 60 SIN 90

.2588 .7071 1 .1736 .1736 .0875 .5 1


Solving Triangles

Carefully analyze what given information you have.

Decide what you are trying to find. Ask: how do you use the given

information to solve the problem? WRITE AN EQUATION. (SOHCAHTOA)

Solve.


Example 1 Find x.

28

x 15

From the 28 angle, x is the ?

Opposite side,

and 15 is the

Hypotenuse.

What trig ratio is this?

Sine (SOHCAHTOA)


Example 1 Find x.

sin 2815

15sin 28

15(.4695)

7.0

x

x

x

x

28

x 15

Write the equation and solve.


Example 2 Find y.

cos3156

56cos31

56(.8572)

48.0

y

y

y

y

31y

56



Example 3 Find a.

tan 408

8tan 40

8(.8391)

6.7

a

a

a

a

40

a

8



Algebra Reminder

Solve: 123

12 3

12 3

3 34

xx

x

x

a ac b

b c


Example 4 Find a.

17tan 40

tan 40 17

17 17

tan 40 .839120.3

aa

a

x

40a

17



Example 5 Find x & y.

tan 78150

150 tan 78

150(4.7046)

705.7

x

x

x

x

78

x y

150

150cos78

150 150

cos78 .2079721.5

y

y

y


Angle of Elevation

Horizontal

Angle of Elevation


Angle of DepressionHorizontal

Angle of Depression

(.wav)


Problem

30 yd15

h

Standing 30 yards from a tree, the angle of elevation to the top of the tree is 15. How tall is the tree?

tan1530

30 tan15

30(.2679)

8.0

h

h

h

h


Summary

Trig ratios are based on acute angles in right triangles.

They are Sine, Cosine, Tangent. SOHCAHTOA Angle of elevation is from the ground

up.

Geometry 9.5 Trigonometric Ratios May 5, 2015Geometry 9.5 Trigonometric Ratios w/o Calculator2 Goals I can find the sine, cosine, and tangent of an acute.

Documents

adjacent slide

opposite hypotenuse

adjacent hypotenuse

opposite hypotenuse

tangent hypotenuse adjacent

cosine hypotenuse adjacent

sohcahtoa slide

adjacent sine