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MTH 112: Elementary Functions

6.2: Right triangle trigonometry 1/16


Boris IskraDepartment of Mathematics. Oregon State University

Figure: Euclid of Alexandria was a Greek mathematician, often referred to as the“Father of Geometry”

Boris Iskra Department of Mathematics. Oregon State University



6.2: Right triangle trigonometry

Standard labeling for any triangle

A B

C

c

ab

αβ

γ

Vertices: A, B and CAngles: α, β and γ

Sides: a,b and c

a is opposite to α

b is opposite to β

c is opposite to γ




Similar triangles

AB

C

DE

F

G

Similar triangles

Similar triangles have congruent corresponding angles.Corresponding sides of similar triangles are proportionalThey are NOT necessarily the same size




A B

C

θ

adjacent

oppo

site

hypotenuse

Six possible side ratios:

opphyp

,adjhyp

,oppadj

,hypopp

,hypadj

,adjopp




Right triangle definitions of trigonometric functions

Let θ be the acute angle of a right triangle.

A B

C

θ

adjacentoppositehypotenuse

sin(θ) =opphyp

, cos(θ) =adjhyp

, tan(θ) =oppadj

,

csc(θ) =hypopp

, sec(θ) =hypadj

, cot(θ) =adjopp




Example

Example 1

Find the six trigonometric functions for the angle θ , where the lengthof the opposite side is 5 and the length of the hypotenuse is 13.

Solution:By the Pythagorean theorem, the length of the side adjacent to θ is:

adjacent =√

132−52 =√

144 = 12.

sin(θ) =opphyp

=5

13, cos(θ) =

adjhyp

=1213

, tan(θ) =oppadj

=?,

csc(θ) =hypopp

=?, sec(θ) =hypadj

=?, cot(θ) =adjopp

=?




Reciprocal identities (or ratio identities)

csc(θ) =1

sin(θ), sec(θ) =

1cos(θ)

, cot(θ) =1

tan(θ).

Example 2

Find the exact value of: cot(45◦), sec(30◦) and csc(60◦).

1 1

22 x

1

√2

1

60◦ 60◦

30◦30◦

45◦

45◦




Cofunction identities

a

bc

β

α

Write the values of each trigonometricfunction for the angles α and β :

sin(α) =ac= cos(β )

tan(α) =ab= cot(β )

sec(α) =cb= csc(β ).




Some exact values of trigonometric functions

Trigonometry Function TableDegrees Radians

sin(θ) cos(θ) tan(θ) cot(θ) sec(θ) csc(θ)

0◦ 0 0 1 0 undefined 1 undefined

30◦π

612

√3

2

√3

3

√3

2√

33

2

45◦π

4

√2

2

√2

21 1

√2

√2

60◦π

3

√3

212

√3

√3

32

2√

33

90◦π

21 0 undefined 0 undefined 1




α and β are complementary angles

α +β = 90◦

sin(α) = cos(90◦−α) = cos(β ), cos(α) = sin(90◦−α) = sin(β ),

tan(α) = cot(90◦−α) = cot(β ), cot(α) = tan(90◦−α) = tan(β ),

sec(α) = csc(90◦−α) = csc(β ), csc(α) = sec(90◦−α) = sec(β ).

Cofunctions of complementary angles are equal.




Example 3

It is known that tan(

π

8

)=√

2−1. Find tan( 3π

8

).

tan(

3π

8

)= tan

(π

2− π

8

)= cot

(π

8

)=

1tan(

π

8

) = 1√2−1

=√

2+1




Angle of elevation and angle of depression

Many times in application examples involving trigonometry, we considerangles formed by a horizontal line and the line of sight from a referencepoint on the horizontal line to an object below or above it. We refer to suchan angle as:

Angle of elevation if the object is above the horizontal.Angle of depression if the object is below the horizontal.




Some applications

Example 4: Hight of theworld tallest tree

According to the The Guin-ness Book of World Records,the tallest tree currently stand-ing is the Mendocino Tree, acoast redwood at MontgomeryState Reserve near Ukiah, Cal-ifornia. At a distance of 200feet from the base of the tree,the angle of elevation to thetop of the tree is approximately61.5◦. How tall is the tree? 200

61.5◦




200

h

61.5◦

tan(61.5◦) =h

200h = 200tan(61.5◦)= 200∗1.8417≈ 368.35




Example

Example 5: Pinpointing a Fire

A U.S. Forest Service helicopter is flying at a height of 500 feet. Thepilot spots a fire in the distance with an angle of depression of 12◦.Find the horizontal distance to the fire.

500

12◦




Example

500

d

12◦

tan(12◦) =500d

d =500

tan(12◦)

=500

0.2125≈ 2352.315


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