This document provides instruction on right triangles and trigonometric ratios. It begins with examples of finding missing angles and side lengths in right triangles using trigonometric functions like sine, cosine and tangent. Special right triangles involving 30-60-90 and 45-45-90 triangles are discussed. Real world applications include finding the height a skier leaves a ramp and the height of a tree using an angle of elevation. The document also covers cosecant, secant and cotangent functions and has practice problems for students.

1. Holt Algebra 2
UNIT 10.3 RIGHT TRIANGLESUNIT 10.3 RIGHT TRIANGLES
AND TRIGONOMETRIC RATIOSAND TRIGONOMETRIC RATIOS

2. Warm Up
Given the measure of one of the acute
angles in a right triangle, find the
measure of the other acute angle.
1. 45° 2. 60°
3. 24° 4. 38°
45° 30°
66° 52°

3. Warm Up Continued
Find the unknown length for each right
triangle with legs a and b and hypotenuse
c.
5. b = 12, c =13
6. a = 3, b = 3
a = 5

4. Understand and use trigonometric
relationships of acute angles in
triangles.
Determine side lengths of right
triangles by using trigonometric
functions.
Objectives

5. trigonometric function
sine
cosine
tangent
cosecants
secant
cotangent
Vocabulary

6. A trigonometric function is a function whose
rule is given by a trigonometric ratio. A
trigonometric ratio compares the lengths of two
sides of a right triangle. The Greek letter theta θ
is traditionally used to represent the measure of
an acute angle in a right triangle. The values of
trigonometric ratios depend upon θ.

8. The triangle shown at right is
similar to the one in the
table because their
corresponding angles are
congruent. No matter which
triangle is used, the value of
sin θ is the same. The values
of the sine and other
trigonometric functions
depend only on angle θ and
not on the size of the
triangle.

9. Example 1: Finding Trigonometric Ratios
Find the value of the
sine, cosine, and
tangent functions for θ.
sin θ =
cos θ =
tan θ =

10. Check It Out! Example 1
Find the value of the
sine, cosine, and
tangent functions for θ.
sin θ =
cos θ =
tan θ =

11. You will frequently need to determine the value of
trigonometric ratios for 30°,60°, and 45° angles as
you solve trigonometry problems. Recall from
geometry that in a 30°-60°-90° triangle, the
ration of the side lengths is 1: 3 :2, and that in a
45°-45°-90° triangle, the ratio of the side lengths is
1:1: 2.

13. Example 2: Finding Side Lengths of Special Right
Triangles
Use a trigonometric function to find the value of x.
°
x = 37
The sine function
relates the opposite
leg and the
hypotenuse.
Multiply both sides by 74 to solve for x.
Substitute for sin 30°.
Substitute 30° for θ, x for
opp, and 74 for hyp.

14. Check It Out! Example 2
Use a trigonometric function to find the value of x.
The sine function
relates the opposite
leg and the
hypotenuse.
Substitute 45 for θ, x for
opp, and 20 for hyp.
°
°
Substitute for sin 45°.
Multiply both sides by 20 to solve for x.

15. Example 3: Sports Application
In a waterskiing competition,
a jump ramp has the
measurements shown. To
the nearest foot, what is
the height h above water
that a skier leaves the ramp?
5 ≈ h
The height above the water is about 5 ft.
Substitute 15.1° for θ, h for opp., and 19
for hyp.
Multiply both sides by 19.
Use a calculator to simplify.

16. Make sure that your graphing calculator is set to
interpret angle values as degrees. Press .
Check that Degree and not Radian is
highlighted in the third row.
Caution!

17. Check It Out! Example 3
A skateboard ramp will
have a height of 12 in.,
and the angle between
the ramp and the ground
will be 17°. To the nearest inch, what will
be the length l of the ramp?
l ≈ 41
The length of the ramp is about 41 in.
Substitute 17° for θ, l for hyp., and 12
for opp.
Multiply both sides by l and divide by
sin 17°.
Use a calculator to simplify.

18. When an object is above or below another object,
you can find distances indirectly by using the angle
of elevation or the angle of depression between the
objects.

19. Example 4: Geology Application
A biologist whose eye level is 6 ft above the
ground measures the angle of elevation to the
top of a tree to be 38.7°. If the biologist is
standing 180 ft from the tree’s base, what is
the height of the tree to the nearest foot?
Step 1 Draw and label a
diagram to represent the
information given in the
problem.

20. Example 4 Continued
Step 2 Let x represent the height of the tree
compared with the biologist’s eye level.
Determine the value of x.
Use the tangent function.
180(tan 38.7°) = x
Substitute 38.7 for θ, x for opp., and
180 for adj.
Multiply both sides by 180.
144 ≈ x Use a calculator to solve for x.

21. Example 4 Continued
Step 3 Determine the overall height of the
tree.
x + 6 = 144 + 6
= 150
The height of the tree is about 150 ft.

22. Check It Out! Example 4
A surveyor whose eye level is 6 ft above the
ground measures the angle of elevation to the
top of the highest hill on a roller coaster to be
60.7°. If the surveyor is standing 120 ft from
the hill’s base, what is the height of the hill to
the nearest foot?
Step 1 Draw and label a
diagram to represent the
information given in the
problem.
120 ft
60.7°

23. Check It Out! Example 4 Continued
Use the tangent function.
120(tan 60.7°) = x
Substitute 60.7 for θ, x for opp., and
120 for adj.
Multiply both sides by 120.
Step 2 Let x represent the height of the hill
compared with the surveyor’s eye level.
Determine the value of x.
214 ≈ x Use a calculator to solve for x.

24. Check It Out! Example 4 Continued
Step 3 Determine the overall height of the
roller coaster hill.
x + 6 = 214 + 6
= 220
The height of the hill is about 220 ft.

25. The reciprocals of the sine, cosine, and tangent
ratios are also trigonometric ratios. They are
trigonometric functions, cosecant, secant, and
cotangent.

26. Example 5: Finding All Trigonometric Functions
Find the values of the six trigonometric
functions for θ.
Step 1 Find the length of the hypotenuse.
70
24
θ
a2
+ b2
= c2
c2
= 242
+ 702
c2
= 5476
c = 74
Pythagorean Theorem.
Substitute 24 for a and
70 for b.
Simplify.
Solve for c. Eliminate
the negative
solution.

27. Example 5 Continued
Step 2 Find the function values.

28. In each reciprocal pair of trigonometric functions,
there is exactly one “co”
Helpful Hint

29. Find the values of the six trigonometric
functions for θ.
Step 1 Find the length of the hypotenuse.
80
18
θ
a2
+ b2
= c2
c2
= 182
+ 802
c2
= 6724
c = 82
Pythagorean Theorem.
Substitute 18 for a and
80 for b.
Simplify.
Solve for c. Eliminate
the negative
solution.
Check It Out! Example 5

30. Check It Out! Example 5 Continued
Step 2 Find the function values.

31. Lesson Quiz: Part I
Solve each equation. Check your answer.
1. Find the values of the six trigonometric functions
for θ.

32. Lesson Quiz: Part II
2. Use a trigonometric function to find the value
of x.
3. A helicopter’s altitude is 4500 ft, and a plane’s
altitude is 12,000 ft. If the angle of depression
from the plane to the helicopter is 27.6°, what is
the distance between the two, to the nearest
hundred feet?
16,200 ft

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