Introduction to Trigonometry

Basic Trigonometric Functions

What is Trigonometry? The study of triangles Relationship between sides and angles of a right triangle

› What is a right triangle? A triangle with a 90⁰ angle

90°

Review Right Triangles In relation to angle a, the

sides of the triangle are:• hypotenuse - always

longest side and side across from right angle (90⁰)

• adjacent - side closest to

angle a• opposite - side opposite

to angle a

hypotenus

e

a

adja

cent

opposite

90⁰

Review Right Triangles

Label the sides for angle b:

• hypotenuse• adjacent• opposite

b

?

?

?hypotenus

e

opposite

adjacent

90°

Trigonometric FunctionsRatios of the sides in relation to angle a:

sine cosine tangent

hypotenus

e a

adja

cent

opposite

90°

Trigonometric Functions:SINE

abbreviation: sin

sin(a)=› 0 ≤ sin ≤ 1

Example: sin(60°)= = ~.866

Ratio of opposite side to hypotenuse for 60° angle is to 2 (.866 to 1)

hypotenus

e a

adja

cent

opposite

90°

oppositehypotenus

e √32

Trigonometric Functions:COSINE

abbreviation: cos

cos(a)=› 0 ≤ cos ≤ 1

Example: cos(60°)= = .5

Ratio of adjacent side to hypotenuse for 60° angle is 1 to 2 (half)

hypotenus

e a

adja

cent

opposite

90°

adjacenthypotenus

e12

Trigonometric Functions:TANGENTabbreviation: tan

tan(a)=› 0 ≤ tan ≤

Example: tan(60°)= = ~1.732

Ratio of opposite side to adjacent side for 60° angle is to 1 (1.732 to 1)

hypotenus

e a

adja

cent

opposite

90°

oppositeadjacent∞oppositeadjacent

Trigonometric Functions:REMEMBER:

Sine =

Cosine =

Tangent =

hypotenus

e a

adja

cent

opposite

90°

Opposite

AdjacentHypotenuse

Hypotenuse

Adjacent

OppositeSOH – CAH – TOASOH TOACAH

Using Trigonometric Functions:

For any right triangle:calculate other sides if one side and angle known

calculate angle if two sides known

90°

Calculating Sides:One Side and Angle Known

What is known?• angle (50°) and adjacent side (2)

Solving for hypotenuse: Which function uses adjacent and hypotenuse?

hypotenus

e 50°

2

opposite

90°

COSINE

Calculating Sides:One Side and Angle Known

What is known?• angle (50°) and adjacent side (2)

Solving for hypotenuse: cos(50°)= =

hypotenuse = ~3.111

hypotenus

e 50°

2

opposite

90°

2 hypotenuse

3.111

~0.643

Calculating Sides:One Side and Angle Known

Now we know:• angle (50°) and hypotenuse (3.111)

Solving for opposite: Which function uses opposite and hypotenuse?3.111

50°

2

opposite

90°

SINE

Calculating Sides:One Side and Angle Known

Now we know:• angle (50°) and hypotenuse

(3.111)Solving for opposite:

sin(50°)= = opposite = ~2.384

3.111

50°

2

opposite

90°

opposite3.111

2.384

~.766

Calculating Angle:Two Sides KnownWhat is known?• adjacent (3) and opposite (5)

Solving for angle (a): Which function uses

adjacent and opposite?

hypotenus

e a

3

5

90°

TANGENT

Calculating Angle:Two Sides KnownWhat is known?• adjacent (3) and opposite (5)

Solving for angle (a): tan(a)= =* need to use inverse tan → tan-1(.6) = a =

~30.964°

hypotenus

e

3

5

90°

35

a 30.964° .6

TEKS Reference§111.35. Precalculus (c)  Knowledge and skills.(3)  The student uses functions and their properties, tools and technology, to model and solve meaningful problems. The student is expected to:(A)  investigate properties of trigonometric and polynomial functions;