The Six Trig Functions Objective: To define and use the six trig functions.

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Transcript of The Six Trig Functions Objective: To define and use the six trig functions.
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 454590 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 454590 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 306090 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 306090 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 306090 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 467468• 1, 3, 5, 925 odd, 57, 59, 61, 63, 65