Download Algebra and Trig. I 4.3 – Right Angle Trigonometry We construct a

Algebra and Trig. I
4.3 – Right Angle Trigonometry
y
1
P=(x,y)
_
y
P=(x,y)
_
1
y
x
x
x
We construct a right triangle by dropping a line segment from
point P perpendicular to the x-axis.
So now we can view
as the measure of an acute angle in
the right triangle.
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SOHCAHTOA (pronounced so-cah-tow-ah)
Pythagorean Theorem
c
a
θ
b
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Example – Find the value of each of the six trigonometric
functions of θ given the following figure.
c
a=5
θ
b=12
Example – Find the value of each of the six trigonometric
functions of θ given the following figure.
c=3
a=1
θ
b
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Special Triangle Relationships
An equilateral triangle is a triangle with three
equal sides. The three angles of an
equilateral triangle are also equal. Each
angle measures 60°.
An isosceles triangle is a triangle with
exactly two equal sides. The angles opposite
these equal sides are also equal.
A scalene triangle is a triangle with all three
sides unequal.
A 45-45-90 triangle is an isosceles right triangle.
The two base angles are each 45°, and the last angle is 90°.
The sum of the angles of a triangle is 180°.
45
45
90
Properties of a 30-60-90 triangle.
A 30-60-90 triangle is an equilateral triangle cut in half.
An equilateral triangle has angle measures 60-60-60, therefore
when we divide the top angle in half that measure becomes 30°,
the altitude creates a 90° angle at the bottom.
30
2
90
60
1
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So how to find the values of the trigonometric functions at
So how to find the values of the trigonometric functions at
So how to find the values of the trigonometric functions at
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θ
Trigonometric Functions and Complements
Two positive angles are complements if the sum of their angles
is
. For example 70° and 20° are complements because
70°+20°=90°.
The figure to the left shows a
right triangle. Because the sum
90-θ
c
a
of the angles of any triangle is
180°, in a right triangle the sum
of the acute angles is 90°, thus
θ
the acute angles are
b
complements. If one acute
angle is θ° the other must be
90°-θ°
From above we can conclude that
. If two
angles are complements then the sine of one equals the
cosine of the other. Because of this relationship the sine and
cosine functions are called confunctions of each other. (The name
cosine is a shortened form of the phrase complement’s sine.)
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Any pair of trig. functions f and g for which
are called
confunctions.
Confunction Identities –
The value of a trigonometric function of θ is equal to the
confunction of the complement of θ. Confunctions of
complementary angles are equal.
Example – Find a confunction with the same value as the given
expression.
1.
2.
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Applications –
Line of Sight above Observer
Angle of
elevation
Horizontal
Angle of
depression
Line of Sight below Observer
The angle of elevation is the angle from the horizontal line to the
line of sight above the observer
The angle of depression is the angle from the horizontal line to
the line of sight below the observer
Example – A tower that is 125 feet tall casts a shadow 172 feet.
Find the angle of elevation.
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Example – The irregular shape is a lake. The distance across the
lake is unknown. To find the distance a surveyor took the
measurement shown. What is the distance across the lake?
(θ=22°)
a
θ
300 yards
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