Download MATH 175: Chapter 7 Review Analytic Trigonometry

MATH 175: Chapter 7 Review Analytic Trigonometry
In order to prepare for a test on chapter 7, you need to understand and be able to work problems
involving the following topics:
I.
Inverse Trigonometric Functions
A. Can You Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function?
1) Find the exact value of the expression sin
1
3
2
.
2) Find the exact value of the expression cos
1
3
2
.
3) Find the exact value of the expression tan 1 ( 1).
B. Can You Use a Calculator to Find the Value of an Inverse Trigonometric Expression Rounded to Two
Decimal Places?
4) Use a calculator to find the value of the expression sin
1
5) Use a calculator to find the value of the expression cos
1
3
4
rounded to two decimal places.
2
5
rounded to two decimal places.
C. Can You Use Properties of Inverse Functions to Find the Exact Value of a Certain Composite
Function?
6) Find the exact value of the expression cos
1
cos 67 . Do not use a calculator.
7) Find the exact value of the expression tan
1
tan( 6 ) . Do not use a calculator.
8) Find the exact value of the expression sin
1
sin 54 . Do not use a calculator.
D. Given a Trigonometric Function, Can You Find its Inverse and State its Domain and Range?
9) Find the inverse function, f
1
, of the function f x
7 cos x 3 . State its domain and
range.
E. Can You Find the Exact Solution of a Basic Equation Involving Inverse Trigonometric Equations?
10) Find the exact solution of the equation 3sin 1 x
.
F. Can You Express a Trigonometric Expression as an Algebraic Expression?
11) Write the trigonometric expression cos sin 1 u as an algebraic expression in u.
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II. Trigonometric Identities
A. Can You Use Algebra to Simplify Trigonometric Expressions?
12) Multiply and simplify the expression
(tan
1)(tan
tan
1) sec2
.
B. Can You Use the Basic Identities to Establish Other Identities?
13) Establish the identity: (sin x)(tan x cos x - cot x cos x) = 1-2 cos 2 x .
14) Establish the identity.
cos u
cos u sin u
15) Establish the identity. sec u + tan u =
16) Establish the identity.
1
.
1 tan u
cos u
.
1 sin u
sin x
sin x
1 cos x 1 cos x
2csc x .
III. Can You Use the Sum and Difference Formulas to Find Exact Values of Trigonometric
Functions?
17) Find the exact value of the expression sin(
11
12
).
18) Find the exact value of the expression tan 75°.
tan 65o tan 85o
.
19) Find the exact value of the expression
1 tan 65o tan 85 o
20) Find the exact value of the expression cos (5π/18) sin (π/9) - cos (π/9) sin (5π/18).
21) Find the exact value of cos (α + β) under the given conditions.
20
29
sin α =
, 0 < α < (π/2); cos β
12
13
, 0 < β < (π/2).
22) Find the exact value of sin (α + β) under the given conditions.
tan α
15
8
23) If sin θ =
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, π < α < (3π/2); cos β
1
4
24
25
, π/2 < β < π.
, θ in quadrant II, find the exact value of cos(
6
).
2
IV. Can You Use the Sum and Difference Formulas to Establish Identities?
24) Establish the identity cos( x
25) Establish the identity
6
)
cos( x y )
cos( x y )
3
2
cos x
1
2
sin x.
1 tan x tan y
.
1 tan x tan y
V. Can You Use the Sum and Difference Formulas to Evaluate Functions Involving Inverse
Trigonometric Functions?
26) Find the exact value of this expression sin cos
1 1
2
sin
1
3
2
.
27) Write the trigonometric expression cos sin 1 u cos 1 v as an algebraic expression containing
u and v.
VI. Can You Use Double-angle and Half-angle Formulas to Evaluate Trigonometric Functions
and Prove Identities?
A. Use Double-angle Formulas to Find Exact Values.
28) Find cos(2θ) given that sin θ =
15
17
, 0 < θ < π/2.
29) Find cos(2θ) given that cos θ =
1
3
, csc θ < 0.
30) Find sin(2θ) given that sin θ =
4
5
, 3π/2 < θ < 2π.
31) Find the exact value of the expression sin 2 sin
1
2
2
.
B. Use Double-angle Formulas to Establish Identities.
csc2
.
csc2
2
32) Establish the identity sec(2 )
33) Establish the identity sin(4x) = (4 sin x cos x)( 2cos2 x 1 ).
C. Use Half-angle Formulas to Find Exact Values.
34) Find cos ( 2 ) given that sin θ =
35) Find sin ( 2 ) given that csc θ =
1
4
and tan θ > 0.
6 and cos θ > 0.
36) Find tan ( 2 ) given that tan θ = 3, π < θ < 3π/2.
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VII. Can You Solve Equations Involving Trigonometric Functions?
A. Solve Equations Involving a Single Trigonometric Function.
37) Solve the equation 4sin 2
1 on the interval 0 ≤ θ < 2π.
38) Solve the equation 5 2 sin
4
39) Solve the equation cos(2 )
3
2
40) Solve the equation cos
2
2
1 on the interval 0 ≤ θ < 2π.
on the interval 0 ≤ θ < 2π.
. Give a general formula for all the solutions.
41) Use your graphing utility to solve the equation csc θ = -2 on the interval 0 ≤ θ < 2π. Round the
answer to two decimal places.
42) A mass hangs from a spring which oscillates up and down. The position P (in feet) of the mass at
time t (in seconds) is given by P = 4cos(4t ) . For what values of t, 0 ≤ t < π, will the position be
2 2 feet? Find the exact values. Do not use a calculator.
B. Solve Equations That Are Quadratic in Form.
43) Solve the equation cos2
44) Solve the equation 2sin 2
2cos
3sin
1 0 on the interval 0 ≤ θ < 2π.
2 0 on the interval 0 ≤ θ < 2π.
C. Solve Trigonometric Equations Using Identities.
45) Solve the equation sin 2
5(cos
46) Solve the equation cos(2 ) 6sin 2
1) on the interval 0 ≤ θ < 2π.
2 on the interval 0 ≤ θ < 2π.
47) Solve the equation sin(2θ) + sin θ = 0 on the interval 0 ≤ θ < 2π.
D. Use a Calculator to Solve Equations That Include Trigonometric Functions.
48) Use your graphing utility to solve the equation 6x - 5 sin x = 2. Round the solution(s) to two
decimal places if necessary.
49) Use a graphing utility to solve the equation 11 24sin x 16cos2 x on the interval
0° ≤ x < 360°. Express the solution(s) rounded to one decimal place.
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MATH 175 Chapter 7 Review: Analytic Trigonometry
Answer Key
1)
2)
3)
4)
5)
6)
7)
8)
9)
π/3
π/6
-π/4
-0.85
1.86
6π/7
- π/6
- π/4
1
f
10) x =
cos 1
x
x 3
. Domain:
7
4, 10 . Range: 0,
3
2
1 u2
2 cot
11)
12)
2
13) sin x(tan x cos x cot x cos x) sin x sin x cos x cos x
cos x
14)
cos u
cos u sin u
15) sec u tan u
16)
1
cos u
sin x
sin x
1 cos x 1 cos x
2
17)
sin u
cos u
1
cos u
1
cos u
1
1
1 sin u
cos u
sin u
cos u
(1 cos 2 x) cos 2 x 1 2 cos 2 x.
1
1 tan u
1 sin 2 u
cos u (1 sin u )
1 sin u 1 sin u
cos u 1 sin u
sin x[1 cos x 1 cos x]
(1 cos x)(1 cos x)
2sin x
1 cos 2 x
2sin x
sin 2 x
cos 2 u
cos u(1 sin u)
cos u
.
1 sin u
2csc x.
6
4
18) 2
3
3
19)
3
20)
1
2
21)
152
377
22)
304
425
23)
cos u
cos u sin u
sin 2 x cos 2 x
sin x
3 5 1
8
24) cos (x + (π/6)) = cos x cos (π/6) - sin x sin (π/6) =
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3
2
cos x
1
2
sin x .
5
cos( x y )
cos( x y )
26) 0
cos x cos y sin x sin y
cos x cos y sin x sin y
25)
2
27) v 1 u
1/(cos x cos y ) cos x cos y sin x sin y
1/(cos x cos y ) cos x cos y sin x sin y
1 tan x tan y
1 tan x tan y
u 1 v2
161
289
7
29)
9
24
30)
25
28)
31) 1
32) sec(2 )
1
cos(2 )
1
1 2sin 2
1
sin 2
1
sin 2
2
csc 2
csc 2
2
33) sin(4x) = 2 sin(2x) cos(2x) = (4 sin x cos x)(2 cos 2 x - 1).
4
34)
15
8
6
35)
30
12
1
36)
10
5
6
3
, 76 , 116
37)
6
38)
5
4
, 74
39)
12
, 1112 , 1312 , 2312
,
40) {θ| θ = (3π/4) + 2kπ, θ = (5π/4) + 2kπ} where k = 0, 1, 2,.......
41) 5.76, 3.67
42) t
16
, 716 , 916 , 1516
43) π
44)
7
6
, 116
45) π
46)
6
, 56 , 76 , 116
47) 0, 23 , , 43
48) 1.06
49) 48.6°, 131.4°
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