right triangle trigonometry

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Right Triangle Trigonometry

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Take a look at the right triangle, with an acute angle, , in the figure below.

Notice how the three sides are labeled in reference to .

The sides of a right triangle

Side adjacent to

S

ide

op

po

site

Hypotenuse

In this section, we will be studying special ratios of the sides of a right triangle, with respect to angle, .

These ratios are better known as oursix basic trig functions:

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Definitions of the Six Trigonometric Functions

tan

co

=

=t

sin cos

csc sec

opposite adjacent opposite

hypotenuse hypotenuse adjacent

hypotenuse hypotenuse adjacent

opposite adjacent opposite

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To remember the definitions of sine, cosine and tangent, we use the acronym :

“SOH CAH TOA”

Definitions of the Six Trigonometric Functions

O A O

H HS C

AT

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Find the exact value of the six trig functions of in the triangle below:

Example

First find the length of the hypotenuse using the Pythagorean Theorem.

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9

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Given that is an acute angle and , find the

exact value of the five remaining trig functions of .

Example12

cos13

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Find the value of sin given cot = 0.387, where is an

acute angle. Give answer to three significant digits. (Divide out ratio.)

Example

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The 45º- 45º- 90º Triangle

Special Right Triangles

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45º

45º

Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 45

sin 45 = csc 45 =

cos 45 = sec 45 =

tan 45 = cot 45 =

Ratio of the sides:

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The 30º- 60º- 90º Triangle

Special Right Triangles

Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 30

sin 30 = csc 30 =

cos 30 = sec 30 =

tan 30 = cot 30 =

1

3

60º

30º

2

Ratio of the sides:

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The 30º- 60º- 90º Triangle

Special Right Triangles

Find the exact values & decimal approximations (to 3 sig digits) of the six trig functions for 60

sin 60 = csc 60 =

cos 60 = sec 60 =

tan 60 = cot 60 =

1

3

60º

30º

2

Ratio of the sides:

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Using the Calculator to Evaluate Trig Functions

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To evaluate trig functions of acute angles other than 30, 45, and 60, you will use the calculator.

Your calculator has keys marked , , and .

**Make sure the MODE is set to the correct unit of angle measure. (Degree vs. Radian)

Example:

Find to two decimal places.

tan 46.2

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Angles and Accuracy of Trigonometric Functions

Measurement of Angle to Nearest

Accuracy of Trig Function

1° 2 significant digits

0. 1° or 10' 3 significant digits

0. 01° or 1'4 significant digits

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For reciprocal functions, you may use the button, but DO NOT USE THE INVERSE FUNCTIONS (e.g. SIN-1 )!

Example:

1. Find 2. Find

(to 3 significant dig) (to 4 significant dig)

Using the Calculator to Evaluate Trig Functions

csc73.2 cot 11.56

To find the values of the remaining three functions (cosecant, secant, and tangent), use the reciprocal identities.

X

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The inverse trig functions give the measure of the angle if we know the value of the function.

Notation:The inverse sine function is denoted as sin-1x or arcsin x.

It means “the angle whose sine is x”.

The inverse cosine function is denoted as cos-1x or arccos x. It means “the angle whose cosine is x”.

The inverse tangent function is denoted as tan-1x or arctan x. It means “the angle whose tangent is x”.

The Inverse Trigonometric Functions

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Example of Inverse Trig Function

For example, will yield the acute angle whose sine is .

You can think of this as the related equation

1 1sin2

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sin2

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Examples using common angles

Evaluate the following inverse trig functions using the special triangles (you do not need a calculator):

1) 2) 1tan 3 1 1cos

2

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Examples using the calculator

Evaluate the following inverse trig functions using the

calculator. Give answer in degrees.

1 11. tan 1.372 2. sin 0.64

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Examples using the calculator

Evaluate the following inverse trig functions using the

calculator. Give answer in degrees.

1 13. tan 1 4. cos 0.541

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Using Trig Ratios to Find Missing Parts of Right Triangles

Example:

Solve for y in the right triangle below:

Solution:

y

52º

9.6Since you are looking for the side adjacent to 52º and are given the hypotenuse, you could use the _____________ function.

Solving Right TrianglesTo solve a right triangle is to find any missing angles and

any missing sides.

• You will always be given 3 parts, and you will need to find 3 parts.

• The angles are labeled using capital letters A, B, & C. Use angle C to represent the right angle. Angles A and B represent the acute angles.

• The sides are labeled using lowercase letters a, b, & c. Each side is labeled with respect to its opposite angle.

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Example

Solve the right triangle with the indicated measures.

1. 40.7 8.20A a in

Solution

A= 40.7°

C B

b c

a=8.20”

Answers:

B

b

c

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Example

2. 25.8 35.4a c A

C B

b c=35.4

a=25.8

A

B

b

Answers:

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Example

3. Find the altitude of the isosceles triangle below.

36°

8.6 m

36°

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Example

4. Solve the right triangle with 8.600 11.25a cm b cm

Answers:

A

B

c

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Angle of Elevation and Angle of Depression

The angle of elevation for a point above a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point.

The angle of depression for a point below a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point.

Horizontal line

Horizontal line

Angle of elevation

Angle of depression

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Example

A guy wire of length 108 meters runs from the top of an

antenna to the ground. If the angle of elevation of the top

of the antenna, sighting along the guy wire, is 42.3° then

what is the height of the antenna? Give answer to three

significant digits.

Solution

108 m

42.3°

y

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