Trigonometry

Trigonometry

Right Triangles Non-Right Triangles

1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot2. a2 + b2 = c2

3. Radian Measure of angles4. Unit circle5. Inverse trig functions

1. Exact values

3. Changing units.

5. Calculator work

1. Law of Sines2. Law of Cosines

: AAS, ASA, SSA: SAS, SSS

Right Triangles“naming the sides of the triangle.”

What we call the

legs of the triangle

depend on the non-

right angle given.

hypotenuse

opposite

adjacentopposite

adjacentThis is important because all of the trig functions are ratios that are defined by

the lengths of these sides. For example: sine of an angle is the ratio of

the length of the side opposite the angle divided by the length of the

hypotenuse.

Sin θ = hypopp

Confused?

hypotenuse

opposite

adjacent

hypotenuseopposite

adjacent

Trig FunctionsThere are 6 trig functions we

must be able to use. We must memorize their EXACT values in both radical and radian form. Remember: trig functions are

the result of ratios of the lengths of sides of a right triangle.

Trig FunctionsThere are 3 main trig functions and the 3 that are reciprocals of

the first three. The main ones are: Sine, Cosine

and Tangent.sin θ = opp hyp

cos θ = adj hyp

tan θ = opp adj

The reciprocals are: Cosecant, Secant and Cotangent.

csc θ = hyp opp

sec θ = hyp adj

cot θ = adj opp

Basically, to find the trig

relationship of any angle on a

right triangle, all we need to do is

measure the appropriate sides of that triangle.

This is called “evaluating the trig

functions of an angle θ.”

hypotenuseopposite

adjacent

Evaluate the six trig functions of the angle θ.

θ

sin θ = opp hyp

3

4

5

sin θ = 3 5

cos θ = adj hyp cos θ = 4 5

tan θ = opp adjtan θ = 3 4

csc θ = 5 3

sec θ = 5 4

cot θ = 4 3

hypotenuse

opposite

adjacent

We can work backwards as well. If they give us the ratio, we can find the other trig

functions.θ

sin θ = opp hyp

Given: sin θ = 5 6

cos θ = 6

tan θ = 5

sec θ = 6

csc θ = 6 5

cot θ = 5

5

6a2 + b2 = c2

11

1111

11

11

sec θ = 6 11

11

tan θ = 5 11

11

Special Triangles: 30-60-90 and 45-45-90

30˚

60˚

45˚

45˚1

1

1

2 3 2

30˚

60˚

45˚

θ sin θ cos θ

tan θ

csc θ

sec θ

tan θ

30˚

60˚

45˚

θ sin θ cos θ

tan θ

csc θ

sec θ

tan θ

Find the exact values of x and y.

60˚x

8 y

Find the values of x and y.

35˚y

x 16

This is 1 unit long.

180˚ = π radians

360˚ = 2π radians

Hence the name:

The UNIT CIRCLE

90˚ = π radians 2

Since 180 ˚ = 1π radians we can us this as our conversion factor.

In other words to change

degrees into radian we multiply by π

180˚

To change

radians into degrees we multiply by

π

180˚

Hint: What we “want” is always in the numerator. If we want our final answer in degrees then 180 ˚ is on

top. If we want radians then π radians in on top!

Convert 230˚ to radians.

Since we want radians we multiply by π/18

(radians in the numerator.

230˚● π = 230π180˚ 180

Which reduces to 23π 18

NO MIXED FRACTIONS!!!

Convert π to degrees12

Since we want degrees we multiply by 180/π

(degrees in the numerator.)Notice the π’s

cancel!

180

12

Reduces to 15˚

●

●●

●(4, 12)(4, 12)

adjacent

opposite

hypo

tenu

se

radi

us

This leads us to believe that there must be a connection between

sin, cos and the coordinates (x, y)

The UNIT CIRCLE

Remember the unit circle has a radius of 1 unit.

●

So to find the coordinates of this point we can use the sin and cos if we know what the measure of the

angle formed by the radius and the x axis is..

θ

( the length of the pink line, the length of the red line)

BUT WAIT! That’s what cos and sin are defined as!

sinθ = length of side opposite length of hypotenuse

cosθ = length of side adjacent length of hypotenuse

AND WE KNOW THAT THE RADIUS IN A UNIT CIRCLE IS 1 so that means:

sinθ = length of side opposite

cosθ = length of side adjacent

( cos θ, sin θ )

What are the coordinates

of ●

θ

( the length of the pink line, the length of the red line)

( cos θ, sin θ )

The UNIT CIRCLE

●

●●

●(4, 12)(4, 12)

adjacent

opposite

hypo

tenu

se

radi

us