1. Right Triangle Trigonometry
Chapter 13 Sections

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13.1
Trigonometric Ratios
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3. Trigonometric Ratios
Many applications in science and technology require the use of triangles and trigonometry.
These ratios apply only to right triangles. A right triangle has one right angle, two acute angles, a hypotenuse, and two legs.
The right angle, as shown in
Figure 13.1, is usually labeled
with the capital letter C.
Right triangle
Figure 13.1

4. Trigonometric Ratios
The vertices of the two acute angles are usually labeled with the capital letters A and B.
The hypotenuse is the side opposite the right angle, the longest side of a right triangle, and is usually labeled with the lowercase letter c.
The legs are the sides opposite the acute angles.
The leg (side) opposite angle A is labeled a, and the leg opposite angle B is labeled b.

5. Trigonometric Ratios
Note that each side of the triangle is labeled with the lowercase of the letter of the angle opposite that side.
The two legs are also named as the side opposite angle A and the side adjacent to (or next to) angle A or as the side opposite angle B and the side adjacent to angle B.
Key parts of a right triangle

6. Trigonometric Ratios Pythagorean Theorem In any right triangle,
c2 = a2 + b2
That is, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

7. Trigonometric Ratios
The following equivalent formulas are often more useful:
used to find the length of the hypotenuse
used to find the length of leg a
used to find the length of leg b

8. Example 1 Find the length of side b in Figure 13.3.
Using the formula to find the length of leg b, we have
Figure 13.3

9. Trigonometric Ratios
A ratio is the comparison of two quantities by division.
The ratios of the sides of a right triangle can be used to find an unknown part—or parts—of that right triangle.
Such a ratio is called a trigonometric ratio and expresses the relationship between an acute angle and the lengths of two of the sides of a right triangle.

10. Trigonometric Ratios
The sine of angle A, abbreviated “sin A,” equals the ratio of the length of the side opposite angle A, which is a, to the length of the hypotenuse, c.
The cosine of angle A, abbreviated “cos A,” equals the ratio of the length of the side adjacent to angle A, which is b, to the length of the hypotenuse, c.
The tangent of angle A, abbreviated “tan A,” equals the ratio of the length of the side opposite angle A, which is a, to the length of the side adjacent to angle A, which is b.
Note: We could just as easily replace angle A above with angle B but be careful since the sides opposite and adjacent will switch.

11. Trigonometric Ratios
That is, in a right triangle (Figure 13.5), we have the following ratios.
S C T
Trigonometric Ratios O A O
H H A
S (sine)
O (opposite side)
H (hypotenuse)
C (cosine)
A (adjacent)
T (tangent)
O (opposite)
Figure 13.5

12. Trigonometric Ratios Note
When working with the trigonometric functions on your calculator, make certain that it is set in the degree mode.
Most scientific calculators are preset to normally be in this mode and you will see
“deg” up in the top line of window.

13. Example 3
Find the three trigonometric ratios for angle A in the triangle in Figure 13.6.
Figure 13.6

14. Example 3
cont’d

15. Example 3
cont’d

16. 13.2 Using Trigonometric Ratios to Find Angles
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17. Example 1 In Figure 13.7, find angle A using a calculator, as follows.
We know the sides opposite and adjacent to angle A. So we use the tangent ratio:
We can now use the SHIFT and SIN
keys on the calculator to tell us the < of this ratio
would be degrees. Round to 53.2 since the sides
given above are rounded to the nearest tenth. So
< A = 53.2
Figure 13.7

18. Using Trigonometric Ratios to Find Angles
Which Trig Ratio to Use
If you are finding an angle, two sides must be known.
Label these two known sides as side opposite the angle you are finding, side adjacent to the angle you are finding, or hypotenuse.
Then choose the trig ratio that has these two sides.

19. Using Trigonometric Ratios to Find Angles
A useful and time-saving fact about right triangles is that the sum of the acute angles of any right triangle is 90°.
We know that the sum of the interior angles of any triangle is 180°. A right triangle, by definition, contains a right angle, whose measure is 90°. That leaves 90° to be divided between the two acute angles.
Figure 13.10

20. Using Trigonometric Ratios to Find Angles
Note, then, that if one acute angle is given or known, the other acute angle may be found by subtracting the known angle from 90°. That is,

21. Example 3 Find angle A and angle B in the triangle in Figure 13.11.

22. Example 3
cont’d
Use the SHIFT TAN buttons on your calculator to find < A

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13.3
Using Trigonometric Ratios to Find Sides
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24. Using Trigonometric Ratios to Find Sides
Which Trig Ratio to Use
If you are finding a side, one side and one angle must be known.
Label the known side and the unknown side as side opposite the known angle, side adjacent to the known angle, or hypotenuse.
Then choose the trig ratio that has these two sides.

25. Example 1 Find side a in the triangle in Figure 13.12.
With respect to the known angle B, we know the hypotenuse and are finding the adjacent side.
Figure 13.12

26. Example 1 So we use the cosine ratio.
cont’d
So we use the cosine ratio.
(Think of cos 24.5 as a fraction or ratio
using 1 as the denominator)
so cos = __a__ ft
(1)
Cross multiply.

27. Example 1 Side a can be found by using a calculator as follows.
cont’d
Side a can be found by using a calculator as follows.
24.5 COS X =
Thus, side a = 235 ft rounded to the nearest foot.

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13.4
Solving Right Triangles
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29. Solving Right Triangles
The phrase solving a right triangle refers to finding the measures of the various parts of a triangle that are not given.
In other words, to solve a right triangle you need to find both acute angles and all three sides.

30. Example 1 Solve the right triangle in Figure 13.14.
We are given the measure of one acute angle and the length of one leg.
A = 90 – B
A = 90° – 36.7°
= 53.3°
Figure 13.14

31. Example 1
cont’d
We then can use either the sine or the cosine ratio to find side c.
Think sin 36.7 / 1 for the ratio on the left.
Cross multiply
Divide both sides by sin 36.7°.

32. Example 1
cont’d
Now we may use either a trigonometric ratio or the Pythagorean theorem to find side a.
Solution by a Trigonometric Ratio:
Think of tan 36.7 / 1 for the ratio on the left!
Multiply both sides by a.

33. Example 1 Solution by the Pythagorean Theorem: cont’d
Divide both sides by tan 36.7°.

34. 13.5 Applications Involving Trigonometric Ratios
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35. Example 1
The roof in Figure has a rise of 7.50 ft and a run of 18.0 ft. Find angle A.
We know the length of the side opposite angle A and the length of the side adjacent to angle A.
Figure 13.17

36. Example 1
cont’d
So we use the tangent ratio.

37. Applications Involving Trigonometric Ratios
The angle of depression is the angle between the horizontal and the line of sight to an object that is below the horizontal. The angle of elevation is the angle between the horizontal and the line of sight to an object that is above the horizontal.

38. Applications Involving Trigonometric Ratios
In Figure 13.18, angle A is the angle of depression for an observer in the helicopter sighting down to the building on the ground, and angle B is the angle of elevation for an observer in the building sighting up to the helicopter.
Figure 13.18

39. Example 2
A ship’s navigator measures the angle of elevation to the beacon of a lighthouse to be 10.1°. He knows that this particular beacon is 225 m above sea level. How far is the ship from the lighthouse?
First, you should sketch the problem, as in Figure
Figure 13.19

40. Example 2
cont’d
Since this problem involves finding the length of the side adjacent to an angle when the opposite side is known, we use the tangent ratio.
Multiply both sides by b.
Divide both sides by tan 10.1.
b = 1263