Download 181 7–3 Finding the Angle When the Trigonometric Function Is Given

Section 7–3
◆
Finding the Angle When the Trigonometric Function Is Given
measured distances to write the six trigonometric ratios of the angle to two decimal places.
Check your answer by calculator.
1. 30
3. 14
2. 55
4. 37
Plot the given point on coordinate axes, and connect it to the origin. Measure the angle formed
with a protractor. Check by taking the tangent of your measured angle. It should equal the ratio y/x.
5. (3, 4)
6. (5, 3)
7. (1, 2)
9. Write the six trigonometric ratios for angle in Fig. 7–9.
8. (5, 4)
122
196
FIGURE 7–9
Each of the given points is on the terminal side of an angle. Compute the distance r from the
origin to the point, and write the sin, cos, and tan of the angle. Work to three significant digits.
10. (3, 5)
12. (5, 3)
14. (4, 5)
11. (4, 2)
13. (2, 3)
15. (4.75, 2.68)
Evaluate. Work to four decimal places.
16. sin 47.3
19. cot 18.2
22. cos 86.75
17. tan 44.4
20. csc 37.5
23. sec 12.3
18. sec 29.5
21. cos 21.27
24. csc 12.67
Write the sin, cos, and tan for each angle. Keep three decimal places.
25. 72.8
27. 33.1
29. 0.744 rad
26. 19.2
28. 41.4
30. 0.385 rad
Evaluate each trigonometric expression to three significant digits.
31. 5.27 sin 45.8 1.73
33. 3.72(sin 28.3 cos 72.3)
35. 2.84(5.28 cos 2 2.82) 3.35
7–3
32. 2.84 cos 73.4 3.83 tan 36.2
34. 11.2 tan 5 15.3 cos 3
36. 2.63 sin 2.4 1.36 cos 3.5 3.13 tan 2.5
Finding the Angle When the Trigonometric Function Is Given
The operation of finding the angle when the trigonometric function is given is the inverse of
finding the function when the angle is given. There is special notation to indicate the inverse
trigonometric function. If
sin A
181
182
Chapter 7
◆
Right Triangles and Vectors
we write
arcsin A
or
sin
1 A
which is read “ is the angle whose sine is A.” Similarly, we use the symbols arccos A, cos
1 A,
arctan A, and so on. Some calculators use INV SIN A, for example.
Common
Error
◆◆◆
Do not confuse the inverse with the reciprocal.
1
(sin )
1 sin
1 sin Example 12: If sin 0.7337, find in degrees to three significant digits.
Solution:
sin
1 0.7337 47.2
Note that a calculator gives us acute angles only.
◆◆◆
◆◆◆
Example 13: If tan x 2.846, find x in radians to four significant digits.
Solution:
tan
1 2.846 1.233
In this chapter we limit our work
with inverse functions to firstquadrant angles only. In Chapter
15, we consider any angle.
◆◆◆
If the cotangent, secant, or cosecant is given, we first take the reciprocal of that value and
then use the appropriate reciprocal relationship.
1
1
1
sin cos tan csc sec cot ◆◆◆
Example 14: If sec 1.573, find in degrees to four significant digits.
Solution: Since
1
1
cos sec 1.573
we first find the reciprocal of 1.573 and then take the inverse cosine.
1
0.635 73
1.573
Then
cos
1 0.635 73 50.53
Exercise 3
◆
◆◆◆
Finding the Angle When the Trigonometric Function
Is Given
Find the acute angle (in decimal degrees) whose trigonometric function is given. Keep three
significant digits.
1. sin A 0.500
2. tan D 1.53
3. sin G 0.528
4. cot K 1.77
5. sin B 0.483
6. cot E 0.847
Section 7–4
◆
183
Solution of Right Triangles
Without using tables or a calculator, write the sin, cos, and tan of angle A. Leave your answer
in fractional form.
3
12
12
7. sin A 8. cot A 9. cos A 13
5
5
Evaluate the following, giving your answer in decimal degrees to three significant digits.
10. arcsin 0.635
13. cot
1 1.17
7–4
12. tan
1 2.85
15. arccsc 4.26
11. arcsec 3.86
14. cos
1 0.229
Solution of Right Triangles
Right Triangles
The right triangle (Fig. 7–10) was introduced in Sec. 6–2, where we also introduced
the Pythagorean theorem (Eq. 145) and the equation for the sum of the interior angles
(Eq. 139).
Since the angle C is always 90 for right triangles, the equation for the sum of the
interior angles can be re-written as
A B 90 180
A
c
b
C
or
A B 90
This plus our trigonometric relations for sin, cos, and tan become our tools for solving
any right triangle.
Pythagorean
Theorem
c2 a2 b2
145
Sum of the
Angles
A B 90
139
Trigonometric
Functions
side opposite to sin hypotenuse
146
side adjacent to cos hypotenuse
147
side opposite to tan side adjacent to 148
Solving Right Triangles When One Side and One Angle Are Known
To solve a triangle means to find all missing sides and angles (although in most practical problems we need find only one missing side or angle). We can solve any right triangle if we know
one side and either another side or one angle.
To solve a right triangle when one side and one angle are known:
1. Make a sketch.
2. Find the missing angle by using Eq. 139.
a
B
FIGURE 7–10 A right triangle. We will usually label a right
triangle as shown here. We label
the angles with capital letters
A, B, and C, with C always the
right angle. We label the sides
with lowercase letters a, b, and
c, with side a opposite angle
A, side b opposite angle B, and
side c (the hypotenuse) opposite
angle C (the right angle).