The Six Trig Functions

Objective: To define and use the six trig functions

SOH CAH TOA

• We will start by looking at three of the trig functions: the sine, cosine, and tangent. We will use a right triangle to help define them.

hypotenuse

oppositesin

hypotenuse

adjacentcos

adjacent

oppositetan

SOH CAH TOA

• Each of these functions has a reciprocal function. The cosecant goes with the sine, the secant goes with the cosine, and the cotangent goes with the tangent.

hypotenuse

oppositesin

hypotenuse

adjacentcos

adjacent

oppositetan

opposite

hypotenusecsc

adjacent

hypotenusesec

opposite

adjacentcot

Example 1

• Lets look at the following triangle to find the values of the six trig functions.

Example 1

• Lets look at the following triangle to find the values of the six trig functions.

• The first thing we need to do is use the Pythagorean Theorem to find the hypotenuse.

hypot

hypot

hypotenuse

5

25

432

222

5

Example 1

• Lets look at the following triangle to find the values of the six trig functions.

5

4sin

hypotenuse

opposite

5

3cos

hypotenuse

adjacent

3

4tan

adjacent

opposite

4

5csc

opposite

hypotenuse

3

5sec

adjacent

hypotenuse

4

3cot

opposite

adjacent

5

Example 1

• You Try: Use the following triangle to find the value of all six trig functions.

• Again, you need to use the Pythagorean Theorem to find the missing side.

Example 1

• You Try: Use the following triangle to find the value of all six trig functions.

13

5sin

hypotenuse

opposite

13

12cos

hypotenuse

adjacent

12

5tan

adjacent

opposite

5

13csc

opposite

hypotenuse

12

13sec

adjacent

hypotenuse

5

12cot

opposite

adjacent

12

Example 2

• We will use our knowledge of a 45-45-90 triangle to find the sin, cos, and tan of a 450 angle. We will look at this on a unit circle (a circle with a radius of 1).

Example 2

• We will use our knowledge of a 45-45-90 triangle to find the sin, cos, and tan of a 450 angle. We will look at this on a unit circle (a circle with a radius of 1).

2

2

2

145sin 0

hypotenuse

opposite

2

2

2

145cos 0

hypotenuse

adjacent

11

145tan 0

adjacent

opposite

Example 3

• We will use our knowledge of a 30-60-90 triangle to find the sin, cos, and tan of a 300 and 600 angles. We will look at this on a unit circle.

Example 3

• We will use our knowledge of a 30-60-90 triangle to find the sin, cos, and tan of a 300 and 600 angles. We will look at this on a unit circle.

2

360sin 0

hypotenuse

opposite

2

160cos 0

hypotenuse

adjacent

31

360tan 0

adjacent

opposite

Example 3

• We will use our knowledge of a 30-60-90 triangle to find the sin, cos, and tan of a 300 and 600 angles. We will look at this on a unit circle.

2

130sin 0

hypotenuse

opposite

2

330cos 0

hypotenuse

adjacent

3

3

3

130tan 0

adjacent

opposite

Trig Identities

• We stated earlier that some functions were reciprocals of others. We will now look at them this way:

csc

1sin

sec

1cos

cot

1tan

sin

1csc

cos

1sec

tan

1cot

Trig Identities

• There are two more important identities that you need to know.

cos

sintan

1cossin 22

The Calculator

• Your calculator only has the sin, cos, and tan functions on it. To find the other three, you need to use your reciprocal relationships.

• Always make sure that when you are working in radians, your calculator is in radians.

The Calculator

• Your calculator only has the sin, cos, and tan functions on it. To find the other three, you need to use your reciprocal relationships.

• Find the

• We will look at this as

8sec

)8/cos(

1

Angle of Elevation/Depression

• From the horizontal, looking up at an object is called the angle of elevation.

• From the horizontal, looking down at an object is called the angle of depression.

Example 7

• A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.30. How tall is the Washington Monument?

Example 7

• A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.30. How tall is the Washington Monument?

feety

y

y

x

y

adj

opp

555

3.78tan115

1153.78tan

3.78tan

0

0

0

Example 8

• An historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle between the bike path and the walkway.

Example 8

• An historic lighthouse is 200 yards from a bike path along the edge of a lake. A walkway to the lighthouse is 400 yards long. Find the acute angle between the bike path and the walkway.

0211 30)(sin

2

1

400

200sin

Example 9

• Find the length of the skateboard ramp.

Example 9

• Find the length of the skateboard ramp.

feetc

c

c

c

7.124.18sin

4

44.18sin

44.18sin

0

0

0

Homework

• Page 467-468• 1, 3, 5, 9-25 odd, 57, 59, 61, 63, 65