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Section 5.3 Notes Page 1
5.3 Computing Values of Trig Functions of Acute Angles
There are certain angles we can put into our trigonometric functions to get exact answers. There are two special
triangles we will begin with. The first one has a 30, 60 and 90 degree angle. The other triangle that has special
values is a triangle with two 45 degree angles and one 90 degree angle.
30 – 60 – 90 Triangle
2x
60
o
x
In this triangle the opposite side of 30 degrees is always half of the
hypotenuse. The adjacent side is always 3 times the opposite. From
this relationship we can get values for 30 and 60 degrees.
30 o
x 3
sin 30 =
x 1
=
2x 2
sin 60 =
x 3
3
=
2x
2
cos 30 =
3
x 3
=
2x
2
cos 60 =
1
x
=
2x 2
tan 60 =
x 3
= 3
x
tan 30 =
x
=
x 3
1
3
=
3
3
45 – 45 – 90 Triangle
45 o
x 2
In this triangle the opposite and adjacent sides are the same. The
hypotenuse is always 2 times the opposite or adjacent. From
this relationship we can get values for 45 degrees.
x
45 o
x
sin 45 =
cos 45 =
tan 45 =
x
x 2
=
1
=
1
x
x 2
x
=1
x
2
2
=
2
2
=
2
2
Section 5.3 Notes Page 2
You will be referring to the following table A LOT, so please bookmark it. This summarizes what we just
covered. I am also including the 90 degree angle at this time.
Table of trigonometric values
θ (degrees) θ (radians) sin θ
cosθ
tan θ
0
0
0
1
0
30
π
1
2
3
2
2
2
1
2
0
3
3
1
6
π
45
2
2
3
2
1
4
π
60
3
π
90
3
undefined
2
EXAMPLE: Find the exact value without using a calculator: 2 cos 2 30 o − sin 30 o
(
This can be rewritten as: 2 cos 30 o
)
2
− sin 30 o . Now substitute values by using the table.
2
⎛ 3⎞
1
⎟ − . Now square everything inside the parenthesis: 2⎛⎜ 3 ⎞⎟ − 1 . Reduce this fraction: 3 − 1 . After
2⎜⎜
⎟
2
2 2
⎝4⎠ 2
⎝ 2 ⎠
subtracting we get 1.
EXAMPLE: Find the exact value without using a calculator:
1 − cos 60 o
.
sin 60 o
1
2 . We can subtract on the top to get:
We can put in values right away from the table:
3
2
1
3
1 2
over the bottom fraction and multiply to get: ⋅
=
.
which is
3
2 3
3
1−
EXAMPLE: Find the exact value without using a calculator: tan
π
4
+ cot
π
4
.
From the table we get a 1 for each trig function, so the problem becomes 1 + 1 = 2.
1
2 . We can flip
3
2
Section 5.3 Notes Page 3
EXAMPLE: Find the exact value without using a calculator: 1 + tan 30 − csc 45
2
o
2
o
2
⎛ 3⎞
1
⎟ −
From using the table we get the following: 1 + ⎜⎜
. After squaring the problem becomes:
2
⎟
⎝ 3 ⎠ ⎛ 2⎞
⎜
⎟
⎜ 2 ⎟
⎝
⎠
1+
1
1
2
3 1
− . After reducing and simplifying we get: 1 + − 2 which is also − 1 = − .
3
3
3
9 2
4
EXAMPLE: Use trigonometric functions to find x and y:
58
x
30 o
y
Let’s first start with x. We want to choose a trig function that relates the side I want to find with a side that is
given. In our drawing the opposite side is x and the hypotenuse is 58. You want to choose the trig function that
relates these two sides, which is sine. Now we can use the definition of sine to set up our equation:
sin 30 o =
x
58
58 sin 30 o = x
58 ⋅
1
= 29
2
Now we can solve for x by using cross multiplication
From the table we know that sin 30 o =
1
.
2
So we know x = 29.
Now we can solve for y. I will solve this the same way as above. Note you could also use a 2 + b 2 = c 2 fo find
the missing side since we have a right triangle. This time y is the adjacent side and 58 is the hypotenuse. The
trig function that relates these two sides will be cosine. The formula is:
cos 30 o =
y
58
58 cos 30 o = y
58 ⋅
3
= 29 3
2
Cross multiply
From the table we know that cos 30 o =
So we know that y = 29 3 .
3
2
Section 5.3 Notes Page 4
EXAMPLE: A ladder is leaning against a building and forms an angle of 72 degrees with the ground. If the
foot of the ladder is 6 feet from the base of the building how far up the building does the ladder reach? How
long is the ladder?
First let’s draw a picture that describes what is happening. A ladder leaning against a house will give us a right
triangle:
This time we want to solve for y. We can do this the same as in the
previous problem. We want to find a trig function that relates y (opposite)
and 6 (adjacent). This would be tangent. Now we will set up an equation
and solve for y, which is the first thing they are asking us to find.
x
y
72 o
6
tan 72 o =
y
6
Now cross multiply.
6 tan 72 o = y
This time we need to use our calculator. Make sure your calculator is in degree mode.
y = 18.47 feet
You can just round to 2 decimal places.
For the second question we need to solve for x, which will give us the length of the ladder. We need to use
cosine this time since we have an adjacent side and a hypotenuse.
cos 72 o =
6
x
x cos 72 o = 6
x=
6
= 19.42
cos 72 o
Cross multiply.
Divide both sides by cos 72 o
So the length of the ladder is 19.42 feet.
Angle of elevation and depression
When you look up at something you have an angle of elevation, and when you look down on something you
have an angle of depression.
Section 5.3 Notes Page 5
EXAMPLE: While standing 4000 feet away from the base of the CN tower in Toronto, the angle of elevation
was measured to be 24.4 degrees. Find the height of the tower.
First we need a picture. The angle of elevation is 24.4 degrees, so this is as we look up to the top of the tower.
I will call the top of the tower x.
x
24.4
Now we need an equation that relate the opposite side (x) and the adjacent
side (4000). This is tangent. So we can set up an equation.
o
4000
tan 24.4 o =
x
4000
Cross multiply.
4000 tan 24.4 o = x
Make sure your calculator is in degree mode and solve for x.
x = 1814.48 feet
EXAMPLE: An observer in a lighthouse is 66 feet above the surface of the water. The observer sees a ship and
finds the angle of depression to be 0.7 o . Estimate the distance from the ship to the base of the lighthouse in
miles.
We first draw a picture. Notice that the 0.7 degrees is measured from the dotted line since it is an angle of
depression. Remember that angles of elevation and depression are always measured from the HORIZONTAL.
0.7 o
66
0.7 o
From geometry we know that if we have to parallel lines the
alternate interior angles are the same. So this is where the 0.7 o
comes from that is inside the triangle. Now we can use tangent
again since it relates the opposite and adjacent sides.
x
tan 0.7 o =
66
x
x tan 0.7 o = 66
x=
66
= 5401.09 feet
tan 0.7 o
5401.09 ft 1 mile
⋅
= 1.02 miles
1
5280 ft
Cross multiply.
Divide both sides by tan 0.7 o .
This is in feet, so we need to convert it. We know 1 mile = 5280 feet.