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Name ________________________________________ Date __________________ Class __________________
LESSON
13-1
Tangent Ratio
Practice and Problem Solving: A/B
Identify the relationships in the figure to the right.
1. tanX =
3. tan−1
WX
=
VW
= m∠_____
WX
5. tanX × tanV = _____
2. tanV =
4. tan−1
WX
= m∠_____
VW
6. tan−1 VW + tan−1 WX = _____°
WX
VW
Use a calculator to find each tangent or inverse tangent. Round
tangents to the nearest 0.01 and angles to the nearest 0.1 degree.
Check the inverse tangents by finding the tangent of each angle.
7. tan23° ≈ _____________
10. tan−10.14 ≈ _____________°
tan _____________° ≈ 0.14
8. tan43° ≈ _____________
11. tan−11= _____________°
9. tan47° ≈ _____________
12. tan−16.1 ≈ _____________°
tan _____________° = 1
tan _____________° ≈ 6.1
Solve Problems 13–16 using tangent ratios and a calculator. Refer to
the figure to the right of each problem.
13. To the nearest hundredth, what is tanM in
+LMN ? ________
14. Write a ratio that gives tanS. ________ Find the value of tanS to
the nearest hundredth. ________ Use the inverse tangent function
on your calculator to find the angle with that tangent. ________
15. Write and solve a tangent equation to find the distance from
C to E to the nearest 0.1 meter. ________ meters
16. 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. Find EF, the plane’s altitude, to the
nearest foot. Show your work.
_________________________________________________________________________________________
_________________________________________________________________________________________
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Name ________________________________________ Date __________________ Class __________________
LESSON
13-2
Sine and Cosine Ratios
Practice and Problem Solving: A/B
After verifying that the triangle to the right is a right triangle,
use a calculator to find the given measures. Give ratios to the
nearest hundredth and angles to the nearest degree.
1. Use the Pythagorean Theorem to confirm that the triangle
is a right triangle. Show your work.
________________________________________________________________
2. sin∠1 =
≈ _________________
3. sin∠2 =
4. cos∠1 =
= _________________
5. cos∠2 =
= _______________________
≈ _______________________
6. Show how to find m∠1 using the inverse sine of ∠1.
_________________________________________________________________________________________
7. Show how to find m∠2 using the inverse sine of ∠2.
_________________________________________________________________________________________
Use a calculator and trigonometric ratios to find each length.
Round to the nearest hundredth.
8.
9.
BD = _________________
10.
QP = _________________
ST = _________________
Use sine and cosine ratios to solve Problems 11–13.
11. Find the perimeter of the triangle. Round to the nearest
0.1 centimeter. _________________
12. To the nearest 0.1 inch, what is the length of the hypotenuse
of the springboard shown to the right? _________________
13. What is the height of the springboard (the dotted
line)? _________________
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258
Name ________________________________________ Date __________________ Class __________________
LESSON
13-3
Special Right Triangles
Practice and Problem Solving: A/B
Use the figure to the right for Problems 1−4. Write each
trigonometric ratio as a simplified fraction and as a decimal
rounded to the nearest hundredth.
1. sinL
_________________
3. tanM
_________________
2. cosL
_________________
4. sinM
_________________
Write each trigonometric ratio as a simplified fraction.
5. sin 30° = _______________
6. cos 30° = _______________
7. tan 45° = _______________
8. tan 30° = _________________
9. cos 45° = _______________
10. tan 60° = _______________
11. Fill in the side lengths for these special right triangles with a
hypotenuse of 1. Use decimals to the nearest 0.01, and be sure that
your answers make sense, for example that the hypotenuse is longer
than the legs.
Use special right triangle relationships to solve Problems 12–14.
12. If cos A = 0.28, which angle in the triangles to the
right is ∠A? _______________
If sin B = 0.22, which angle is ∠B? _______________
13. What is EF, the measure of the longest side of the sail
on the model? Round to the nearest inch. _________________ in.
What is the measure of the shortest side? _________________ in.
14. If the small sail is similar to the larger one and is 11
inches high, about how wide is it? _________________ in.
Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.
263
Name ________________________________________ Date __________________ Class __________________
LESSON
13-4
Problem Solving with Trigonometry
Practice and Problem Solving: C
For Problems 1–6, use trigonometry and the Pythagorean
theorem to solve the right triangles on the coordinate
plane. Show your work.
1. First use the slope formula to verify that
+ABC is a right
triangle. ____________________________________
2. Use the distance formula to find the length of each side.
AB = ________
BC = ________
AC = ________
3. Use the Pythagorean theorem to double check the side
lengths. ____________________________
4. Use inverse trigonometric ratios to find the acute angles.
m∠A = ________
m∠C = ________
+
5. Verify that PQR is a right triangle. Find the three side
lengths and the measures of the acute angles.
PQ = ________
QR = ________
m∠P = ________
m∠Q = ________
RP = ________
6. Find the side lengths and angle measures for
X(1, 0), Y(2, 1), Z(5, −2).
+XYZ,
XY = ________
YZ = ________
XZ = ________
m∠X = ________
m∠Y = ________
m∠Z = ________
For Problems 7–10, use trigonometric functions to find the
area of the triangles, to the nearest square unit.
7. If you know the lengths of two sides of any triangle, a and b, and
the measure of the included angle, m∠C, how can you find the
area of the triangle? _________________________________
8. Find the area of
+ABC on the coordinate plane above.
_________________________________________________________________________________________
9. Find the area of
+PQR on the coordinate plane above.
_________________________________________________________________________________________
10. Find the area of
+XYZ in Problem 6 above. _________________
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