Download Key Trig Practice Packet

Trigonometry
Many students begin without having been exposed to trigonometry, the mathematics of the
right triangle. This however, is not cause for consternation, as the amount of trig needed for
this course is easily learned. Trig is based on two important facts:
IMPORTANT Fact #1:
The Pythagorean Theorem is true!
a2 + b 2 = c 2
1. Using the Pythagorean Theorem, find the missing side of each right triangle:
a)
b)
c)
d)
269.05 in
11.66 cm
6cm
17.1 ft
12 m
10 m
9.4 ft
97 in
286 in
19.51 ft
10 cm
6.63 m
IMPORTANT Fact # 2: Note that as a triangle grows in size, its angles remain the same and its
sides retain their proportionality. That is, the ratio of any two sides will be unchanged.
48.3
29.1
11.2
26.6o
10
16.8
5
26.6o
7.5
13
26.6 o
26.6 o
15
26
43.2
Now you need to learn some terminology. Study the picture below.
terms are based
on the acute angle
Hypotenuse
Opposite side
you are considering
Adjacent side
See how the angle under consideration is formed by the hypotenuse and an adjacent angle. The
opposite side gets its name from being opposite the angle being considered. Try your new
found knowledge below by identifying the hypotenuse, the adjacent side and the opposite side.
21.6
2. Identify by name the three sides of the triangle. (Note the reference angle)
a)
b)
X=H
Z =A
c)
Y= H
X= O
Y= O
X=A
Y= O
Z= A
Z= H
If you had trouble with finding the hypotenuse, just remember that it’s always the largest side
and is always opposite the right angle.
3. Just to be sure, try one more time on the names of the sides:
a)
b)
Y= A
X=O
c)
Y=H
Z= H
X= H
Y= A
X= A
Z=O
Z= O
Now another new word: The “tangent” of an angle is the opposite side over the adjacent side.
(Keep in mind; we are talking about right triangles only)
tan θ = opp
4. Find the tangents of the angles shown below:
a) 0.75
b) 1.15
19.4 cm
7.0 cm
5.6 cm
θ
2.34 cm
4.2 cm
θ
25.7 cm
adj
c) 0.54
θ
.811 in
16.8 cm
1.12 cm
What about units? Note that since the tangent is a ratio of two lengths, the length
measurements will cancel out, leaving the tangent dimensionless.
5. Use the tangent function on your calculator to find the following:
a) tan 36o
b) tan 71o
0.70
2.90
c) tan 45o
1
d) tan 11o
0.19
6. Use the inverse tangent function to find the angle whose tangent is shown:
a) tan θ = .336
18.57o
b) tan θ = 4.62
77.79o
c) tan θ = 1.44
55.22o
d) tan θ = .0021
0.12o
7. Use the tangent function to find the unknown angles:
a)
b)
10.9 cm
c)
13.5 cm
375 cm
θ
3.39 in
θ
6.7 cm
θ
16.0 cm
94.8 in
9.4 ft
8.6 cm
37.92o
57.47o
49.96o
8. Use the tangent function to find the missing sides:
a)
b)
c)
x
96 ft
x
29
61o
x
55o
o
37.2 m
11.4 in
20.62m
6.32 in
137.10 ft
9. Find the missing angles:
a)
8.9 cm
b)
34.2o
θ
31.6 m
29.1 m
θ
62o
12.3 c m
32.04o
22.63o
There are two more trigonometric functions that will concern us in physics. The “sine” of the
angel is the opposite side divided by the hypotenuse, while the “cosine” of the angle is the
adjacent side divided by the hypotenuse.
10. Find the sine and cosine of each angle below: (abbreviations are sin, cos)
a)
b)
c)
1.41 ft
39.8c m
18.6 cm
θ
30 ft
3.15 ft
θ
660 cm
3.45 ft
35.2 cm
sin θ= 0.47 ,cos θ= 0.88
sin θ = 0.91, cos θ = 0.41
θ
251 in
sin θ = 0.72, cos θ = 0.70
Although the tangent can take on any value from zero to infinity, the sine and cosine are
restricted by their definitions to the range zero to one. This is because the hypotenuse (and
hence the denominator of the fraction) is always larger than either leg.
Negative trigonometric functions occur, but will be left to your trig class.
11. Use your calculator to find the following:
a) sin 15.6o
0.27
b) tan 6.4o
0.11
c) cos 45o
0.71
d) cos 36o
0.81
e) cos 59.4o
0.51
f) sin 30o
0.5
g) sin 1.1o
0.02
h) sin 79.4o
0.98
i) tan 0.633o
0.01
c)
4.89m
12. Use the sine to find the missing side or angle:
a)
b)
60.1 cm
69.51
x ft
32.6 cm
θ
52.6
32.85o
x
73.1o
o
87.5 ft
4.68 m
13. Use the cosine to find the missing side or angle:
a)
11.7m
33.6 ft
b)
c)
o
71.6
x
θ
58.9o
x
551.54cm
289 cm
31 ft
22.69o
3.69 m
With these three functions you can solve any right triangle. The only thing that is tricky for the
novice is determining which trig function to use. Here is the step-by-step approach that will get
you a solution:
1. Identify the right angle.
2. Identify the acute angle and the two sides to be considered. (One of them will be your
unknown).
3. Determine the appropriate trig function (depending on whether you have opp & adj,
opp & hyp, or adj & hyp for your sides).
4. Write out the definition of the function.
5. Plug in your two known values.
6. Solve.
The key to this is being systematic in your approach, and being confident that the solution will
fall into your lap. Here’s your BIG CHANCE:
14. Find the missing side or angle by use of the appropriate function.
a) 10.27ft
b)
x
31.6o
16.7 ft
65.45o
36.5 cm
33.2 cm
x = a/sinθ
c)
θ
θ
a
X=?