Transcript Trigonometry

TRIGONOMETRY
is a branch of Geometry that deals with
TRIANGLES
Why should we learn about Trigonometry?
Trigonometry can be used to
figure out unknown
measurements of parts of
triangles
Figuring out the unknown
measurements of ALL of
the parts of a triangle is
called ‘Solving a Triangle’.
Usually, you can ‘solve a
triangle’ if you know the
measures of three parts of
the triangle.
We will be learning three
applications of trigonometry:
• The three ‘trig functions’:
Sine, Cosine, and Tangent
(abbreviated sin, cos, & tan )
• The ‘Law of Sines’
• The ‘Law of Cosines’
Which of the three applications is to be used depends
on what measurements of a triangle are known:
18
12
16
?
25°
?
45°
14
55°
11
19
23
• The three ‘trig functions’:
(Sine, Cosine, and Tangent)
only apply to RIGHT
triangles
• The ‘Law of Sines’ and the
‘Law of Cosines’ apply to
ALL triangles
Which of the three applications is to be used depends on what
measurements of a triangle are known:
18”
12”
?
Since you know two sides of a right triangle, and you want to
figure out the third side, you use …
Pythagorean Theorem
Which of the three applications is to be used depends on what
measurements of a triangle are known:
25°
16”
?
Trig functions
Since you know three pieces of information ( one is that it is a
right triangle)
Which of the three applications is to be used depends on what
measurements of a triangle are known:
14”
11”
55°
Law of Sines
Since you know one opposite side / angle pair and something else
Which of the three applications is to be used depends on what
measurements of a triangle are known:
115°
19”
23”
x
Law of Cosines
Since you know a ‘complete corner’ : two sides and the
included angle
Which of the three applications is to be used depends on what
measurements of a triangle are known:
xº
23”
19”
28”
Law of Cosines
Since you know three sides and you are trying to find an angle.
The Three Trig functions
• The three 'trig functions' (sine,
cosine, and tangent) are things that
are done to ANGLES, not SIDES.
The important angle of the problem is
called THETA, Θ.
• THETA Θ reminds me of an eyeball,
and that helps me remember that the
angle Θ is the angle that the situation
is viewed from.
Naming the sides of a right triangle
1) Locate the viewpoint, Θ
2) Locate the hypotenuse
3) Locate the adjacent side and the opposite side
hypotenuse
opposite
Θ
adjacent
Naming the sides of a right triangle
1) Locate the viewpoint, Θ
2) Locate the hypotenuse
3) Locate the adjacent side and the opposite side
Θ
hypotenuse
adjacent
opposite
Naming the sides of a right triangle
1) Locate the viewpoint, Θ
2) Locate the hypotenuse
3) Locate the adjacent side and the opposite side
Hypotenuse,…
or opposite ?
adjacent
Opposite,… or another adjacent ?
Θ
The viewpoint is NEVER at the right angle !!
The SINE (abbreviated sin ) is the ratio of
the opposite side divided by the hypotenuse.
20 ”
Θ
12 ”
36.87 º
16 ”
The sine of the 36.87º angle is 12 / 20 , which = 0.6 .
Using your calculator, sin 36.87º = 0.6000.
The COSINE (abbreviated cos ) is the
ratio of the adjacent side divided by the
hypotenuse.
20 ”
Θ
12 ”
36.87 º
16 ”
The cosine of the 36.87º angle is 16 / 20 , which = 0.8 .
Using your calculator, cos 36.87º = 0.7999, = 0.8000.
The TANGENT (abbreviated tan ) is the
ratio of the opposite side divided by the
adjacent side.
20 ”
12 ”
Θ
36.87 º
16 ”
The tangent of the 36.87º angle is 12 / 16 , which = 0.75 .
Using your calculator, tan 36.87º = 0.7500.
So, based upon the location and viewpoint of the
acute angle, called “ Θ ” :
Sin Θ = opposite / hypotenuse
Cos Θ = adjacent / hypotenuse
Tan Θ = opposite / adjacent
Here are two mnemonics to help you remember
these relationships:
•SOH CAH TOA (which sounds like a cool Native
American-ish word )
•Sally Can Tell Oscar Has A Hat On Always
How can you determine the length of the side marked x ?
1) From the viewpoint of the angle Ө, the x side is the opposite side,
Ө
and the 24” side is the hypotenuse.
2) The trig function that relates the opposite
side and the hypotenuse is the sine
67°
24”
x
3) Sin Ө =
opp/
hyp
4) Sin 67º = x / 24
5) 0.9205 = x / 24
22.092”
6) (24) (0.9205) = x
7) (24) (0.9205) = 22.092
How can you determine the measure of angle x ?
1) From the viewpoint of the angle x, the 18” side is the adjacent side,
and the 24” side is the hypotenuse.
2) The trig function that relates the adjacent
side and the hypotenuse is the cosine
24”
41.4 º
x
Ө
3) Cos Ө =
adj/
4) Cos x º =
18/
hyp
24
5) Cos x º = 0.7500
18”
6) What angle has a cosine = 0.7500 ?
7) Cos-1 (0.7500) = 41.40 º
To ‘solve a triangle’ using the three trig functions,
1. Draw and label a picture, including the sides and
the angle. Use a variable to represent the
unknown amount.
2. Identify which two sides are labeled, using the
terms ‘opposite’, ‘adjacent’ and ‘hypotenuse’.
Use the viewpoint of the angle, theta.
3. Determine which of the three trig functions deal
with that pair of side names.
4. Write the proper equation, beginning with the
guide equation. Then, substitute the values you
know for the sides and angles.
5. Solve for the variable using algebra.
For example:
A man in a boat sees the top of the
Barnegat Lighthouse at an angle of 20
degrees above the horizon. He knows
that the lighthouse is 165 feet tall. How
far away from the shore is the boat?
A man in a boat sees the top of the Barnegat Lighthouse at an angle of 20 degrees
above the horizon. He knows that the lighthouse is 165 feet tall. How far away from
the shore is the boat?
This problem models a right
triangle, and we can solve it
using trigonometry !
20°
165'
The question: how long is this side ?
Answer: tan Ө = opp/adj
adj = opp/tan Ө
= 165' / tan 20° = 165' / .364 = 453 feet
THE BOAT IS ABOUT 453 FEET FROM THE SHORE.
In professional football, a net is
raised in front of the bleachers
before a field goal or extra point is
attempted. The net is hung 25 yards
from the end of the end zone.
A groundskeeper on the 30-yard line
sights the top of the net at a 42
degree angle with the ground.
How high is the top of the net?
In professional football, a net is raised in front of the bleachers before a field goal or extra point is
attempted. The net is hung 25 yards from the end of the end zone. A groundskeeper on the 30-yard
line sights the top of the net at a 42 degree angle with the ground. How high is the top of the net?
x
42°
65 yards =
25 +10 +30 yards
In professional football, a net is raised in front of the bleachers before a field goal or extra point is attempted. The net is hung 25 yards
from the end of the end zone. A groundskeeper on the 30-yard line sights the top of the net at a 42 degree angle with the ground. How
high is the top of the net?
x
42°
65 yards
Answer : Tan Θ = opp / adj
Tan 42° = x / 65
.9004 = x / 65
x = (.9004) (65)
x = 58.5
The net is 58.5 yards, or 175.5 feet, above the groundskeeper’s eyes.
So the net is about 181 feet above the ground.
Summary
• Identify the viewpoint angle, Θ
• Use SOH CAH TOA, or some other way to
remember the side ratios.
• Write out the correct trig equation
• Substitute the values for the variables,
and solve for the unknowns using algebra
• If you have to figure out the measure of
the angle, use the inverse trig functions,
sin-1, cos-1, tan-1 (on most calculators, use
the 2nd button)