Chapters 13 – 14 trig

Chapters 13 – 14

Trigonometry

What are Radians?

• Angles can be measured in either degrees

or radians.

• One radian is equal to the measure of a

central angle in a circle whose arc length

equals the radius.

Unit circle

Review of Unit Conversions

• To convert between units of measure:

• Set up a proportion and solve.

• Example: 42 feet is how many yards?

Converting Degrees & Radians

• Fill in the given and cross multiply to solve.

• Example:

• Convert 110 to radians

Examples:

• Convert to degrees.

• Convert to degrees.

Your Turn!

• Convert -220 to radians.

• Convert to degrees.

Special Right Triangles Review

• 45 – 45 – 90

• 30 – 60 – 90

Six Trig Functions

• sin θ = =

• cos θ = =

• tan θ = =

• csc θ = =

• sec θ = =

• cot θ = =

“cosecant”

“secant”

“cotangent”

Evaluating Trig Functions

• Without a calculator!!

1. Find the angle on the unit circle.

2. Evaluate using cosine, sine, or both.

3. Leave answers in reduced radical form.

NO DECIMALS!

Examples

•

• tan 240

•

• csc (-225 )

•

•

Your Turn!

• Evaluate without a calculator:

1. cos (-150 )

2.

Vocab:

• Angles are made of two rays:

▫ The initial side is fixed

▫ The terminal side is rotated about the vertex.

• An angle whose initial side is the + x-axis, and

vertex is the origin is in Standard Position.

General Definition of Trig Functions

• If θ is an angle in standard position,

and (x, y) is a point on the terminal side:

Evaluating Trig Functions

• Let (3, -4) be a point on the terminal side of

an angle θ in standard position. Evaluate the

6 trig functions of θ.

Example

• Let (-5, 12) be a point on the terminal side

of θ. Evaluate the 6 trig functions.

Your Turn!

• Let (-4, -3) be a point on the terminal side of

θ. Evaluate the 6 trig functions of θ.

Modeling with Trig

• A circular clock gear is 2 inches wide. If the

tooth at the farthest right edge starts 10

inches above the base of the clock, how far

above the base is the tooth after it rotates

240 counterclockwise?

Graphing Sine Functions

θ sin θ

0

π/4

π/2

3π/4

π

Graphing Cosine Functions

θ cos θ

0

π/4

π/2

3π/4

π

Vocab:• Cycle – shortest repeating portion.

• Period – horizontal length of each cycle.

• Amplitude – height of the graph, measured

from the center.

Graphing Tangent Functionsθ tan θ

-π

-3π/4

-π/2

-π/4

0

π/4

π/2

3π/4

π

Analyzing Trig Graphs

• Identify the amplitude and period of each:

y = 2 sin x y = 1/2 cos x

Writing Trig Functions• Write an equation for:

• the translation 3 units up of y = sin x.

• the translation π units right of y = cos x.

• the vertical stretch of y = sin x that will double

its amplitude.

• the horizontal stretch of y = cos x that will

double the period.

Your Turn!

• Write an equation of y = sin x after being:

• shifted 3 units down

• shifted π/2 units left

• and vertically compressed to half the original

amplitude.