Trigonometric Ratios
In Exercises 1–3, fill in the blanks to complete each
definition. Then use side lengths from the figure to
complete the indicated trigonometric ratios.
1.The sine (sin) of an angle is the ratio of the length of the leg _____________________
the angle to the length of the _____________________.
sin A 
sin B 
c
2.The cosine (cos) of an angle is the ratio of the length of the leg _____________________
to the angle to the length of the _____________________.
cos A 
cos B 
c
3.The tangent (tan) of an angle is the ratio of the length of the leg _____________________
the angle to the length of the leg _____________________ to the angle.
tan A 
a
tan B 
Use the figure for Exercises 4–6. Write each trigonometric
ratio as a simplified fraction and as a decimal rounded to
the nearest hundredth.
4. sin L
5. cos L
6. tan M
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
7. sin 33
8. cos 47
9. tan 81
Use a calculator and trigonometric ratios to find each length. Round to the
nearest hundredth.
10.
11.
BD ________________
12.
QP _________________
RS _________________
13.
The glide slope is the path a plane uses while it is landing on
a runway. The glide slope usually makes a 3 angle with the
ground. A plane is on the glide slope and is 1 mile (5280 feet)
from touchdown. Use the tangent ratio and a calculator to
find EF, the plane’s altitude, to the nearest foot.
Solving Right Triangles
In Exercises 1–3, fill in the blanks to complete the description of the
inverse trigonometric ratios.
1.If sin A  x, then sin1 x  ________.
2.If cos A  ________, then cos1 x  mA.
3. If tan A  x, then ________  mA.
Use the given trigonometric ratio to determine whether
1 or 2 is A in each exercise.
4. sin A 
4
________
5
5. cos A 
4
________
5
6. tan A 
3
________
4
7. sin A 
3
________
5
8. cos A 
3
________
5
9. tan A 
4
________
3
Use a calculator to find each angle measure to the nearest degree.
10. sin1 (0.33) ________
11. cos1 (0.47) ________
12. tan1 (1.21) ________
 9 
13. sin1   ________
 10 
 1
14. cos1  
5
 3
15. tan1  2  ________
 4
________
Use a calculator and inverse trigonometric ratios to find the unknown
side lengths and angle measures. Round lengths to the nearest
hundredth and angle measures to the nearest degree.
16.
17.
AC 
18.
________
DE 
________
GH 
mB  ________
EF 
________
mH  ________
mC  ________
mD  ________
mI 
XYZ has vertices X(6, 6), Y(6, 3), and Z(1, 3). Complete
Exercises 19–21 to find the side lengths to the nearest
hundredth and the angle measures to the nearest degree.
19. Plot the points and draw XYZ.
20. Tell which angle is the right angle. ________
21. Find XY and YZ from the graph. Use the
Pythagorean Theorem to find XZ.
XY  ________
YZ  ________
XZ  ________
Angles of Elevation and Depression
In Exercises 1 and 2, fill in the blanks to complete the definitions.
1. An angle of elevation is the angle formed by a _______________ line
and a line of sight to a point _______________ the line.
2. An angle of _______________ is the angle formed by a horizontal line and
a line of sight to a point _______________ the line.
________
________
Ben is on the diving board at the neighborhood
pool. Jenna is in the pool, and a lifeguard sits at
her station on the opposite end of the pool.
Classify each angle as an angle of elevation or
an angle of depression.
3. 1 ______________________
4. 2 ______________________
5. 3 ______________________
6. 4 ______________________
Lisa sees a bird’s nest high in a tree. She decides to use
trigonometry to estimate how high the nest is.
7. Lisa walks 15 feet from the base of the tree. She measures an angle
of elevation from the ground to the nest of 62. Find how high the
nest is above the ground, to the nearest foot.
8. Lisa spots the mother bird on a branch above the nest. She
measures an angle of elevation to the bird of 67. Find how
high the mother bird is above the ground, to the nearest foot.
9. Zelda, a trapeze artist, stands on a 10-meter-high platform.
Zelda measures a 40 angle of depression to
the base of the other platform. Find the distance
between the bases of the platforms to the nearest
tenth of a meter
10. Zelda’s partner, Zev, is on the ground doing a safety
check on the net. Zelda measures a 79 angle of
depression to Zev. Find the distance to the nearest
tenth of a meter from Zev to the base of Zelda’s platform.
Law of Sines and Law of Cosines
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
1. sin 168 ________
2. cos 147 ________
3. tan 107 ________
4. sin 97 ________
5. cos 94________
6. tan 140 ________
7. sin 121 ________
8. cos 170 ________
9. tan 135 ________
In Exercises 10 and 11, fill in the blanks to complete the theorems.
10. For any ABC with side lengths a, b, and c,
sin A

a
b

sin C
11. For any ABC with side lengths a, b, and c, a2  b2  c2  2bc cos A,
b2  a2  c2  2ac ________, and ________  a2  b2  2ab cos C.
.
For Exercises 12 and 13, substitute numbers into the given Law of
Sines ratio to find each measure. Round lengths to the nearest tenth
and angle measures to the nearest degree.
12.
13.
sin Q sin R

PR
PQ
sin D sin C

CE
DE
DE _____________________
mR _____________________
Use the Law of Sines to find each measure. Round lengths to the
nearest tenth and angle measures to the nearest degree.
14.
15.
EF _____________________
mN _____________________
For Exercises 16 and 17, substitute numbers into the Law of Cosines to
find each measure. Round lengths to the nearest tenth and angle
measures to the nearest degree.
16.
TU ________
TU 2  ST 2  SU 2  2(ST)(SU)(cos S)
17.
mH ________
GI 2  GH 2  HI 2  2(GH)(HI)(cos H)
Problem Solving – Putting it all together
1. The map shows three earthquake centers for one week in California. How far apart were
the earthquake centers at points
A and C ? Round to the nearest tenth.
2. The coordinates of the vertices of HJK are H(0, 4), J(5, 7), and K(9, 1). Find the
measure of H to the nearest degree.
5. A road has a grade of 28.4%. This means that the road rises 28.4 ft over a horizontal distance
of 100 ft. What angle does the hill make with a horizontal line? Round to the nearest degree.