MATH 175: Chapter 7 Review Analytic Trigonometry - The Learning ...
MATH 175: Chapter 7 Review Analytic Trigonometry - The Learning ...
MATH 175: Chapter 7 Review Analytic Trigonometry - The Learning ...
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<strong>MATH</strong> <strong>175</strong>: <strong>Chapter</strong> 7 <strong>Review</strong> <strong>Analytic</strong> <strong>Trigonometry</strong><br />
In order to prepare for a test on chapter 7, you need to understand and be able to work problems<br />
involving the following topics:<br />
I. Inverse Trigonometric Functions<br />
A. Can You Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function?<br />
1) Find the exact value of the expression<br />
2) Find the exact value of the expression<br />
3) Find the exact value of the expression<br />
1 3<br />
sin<br />
2<br />
.<br />
1 3<br />
cos<br />
2<br />
.<br />
1<br />
tan ( 1).<br />
B. Can You Use a Calculator to Find the Value of an Inverse Trigonometric Expression Rounded to Two<br />
Decimal Places?<br />
4) Use a calculator to find the value of the expression<br />
5) Use a calculator to find the value of the expression<br />
1 3<br />
sin<br />
4<br />
rounded to two decimal places.<br />
1 2<br />
cos<br />
5<br />
rounded to two decimal places.<br />
C. Can You Use Properties of Inverse Functions to Find the Exact Value of a Certain Composite<br />
Function?<br />
6) Find the exact value of the expression cos cos . Do not use a calculator.<br />
1 6<br />
7<br />
1<br />
7) Find the exact value of the expression tan tan( ) . Do not use a calculator.<br />
8) Find the exact value of the expression<br />
6<br />
1 5<br />
sin sin<br />
4<br />
. Do not use a calculator.<br />
D. Given a Trigonometric Function, Can You Find its Inverse and State its Domain and Range?<br />
9) Find the inverse function,<br />
range.<br />
1<br />
f , of the function f x 7cosx<br />
3<br />
. State its domain and<br />
E. Can You Find the Exact Solution of a Basic Equation Involving Inverse Trigonometric Equations?<br />
10) Find the exact solution of the equation<br />
1<br />
3sin x .<br />
F. Can You Express a Trigonometric Expression as an Algebraic Expression?<br />
1<br />
11) Write the trigonometric expression cos sin u as an algebraic expression in u.<br />
REV 12/01/2010 1
II. Trigonometric Identities<br />
A. Can You Use Algebra to Simplify Trigonometric Expressions?<br />
12) Multiply and simplify the expression<br />
(tan 1)(tan 1)<br />
2<br />
sec<br />
tan<br />
.<br />
B. Can You Use the Basic Identities to Establish Other Identities?<br />
2<br />
13) Establish the identity: (sin x)(tan x cos x - cot x cos x) = 1-2 cos x .<br />
14) Establish the identity.<br />
cos u 1<br />
cos u sin u 1 tan u .<br />
cosu<br />
15) Establish the identity. sec u + tan u =<br />
1 sin u .<br />
sin x sin x<br />
16) Establish the identity.<br />
1 cos x 1 cos x<br />
2csc x .<br />
III. Can You Use the Sum and Difference Formulas to Find Exact Values of Trigonometric<br />
Functions?<br />
11<br />
17) Find the exact value of the expression sin(<br />
12<br />
) .<br />
18) Find the exact value of the expression tan 75°.<br />
o<br />
o<br />
tan 65 tan85<br />
19) Find the exact value of the expression<br />
o o .<br />
1 tan 65 tan85<br />
20) Find the exact value of the expression cos (5π/18) sin (π/9) - cos (π/9) sin (5π/18).<br />
21) Find the exact value of cos (α + β) under the given conditions.<br />
sin α =<br />
20<br />
29 , 0 < α < (π/2); cos β 12<br />
13 , 0 < β < (π/2).<br />
22) Find the exact value of sin (α + β) under the given conditions.<br />
15<br />
tan α , π < α < (3π/2); cos β 24<br />
8 25<br />
, π/2 < β < π.<br />
1<br />
23) If sin θ = , θ in quadrant II, find the exact value of 4<br />
cos(<br />
6) .<br />
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IV. Can You Use the Sum and Difference Formulas to Establish Identities?<br />
24) Establish the identity<br />
cos( x ) cos x sin x .<br />
3 1<br />
6 2 2<br />
25) Establish the identity cos( x y ) 1 tan x tan y<br />
.<br />
cos( x y) 1 tan x tan y<br />
V. Can You Use the Sum and Difference Formulas to Evaluate Functions Involving Inverse<br />
Trigonometric Functions?<br />
1 1 3<br />
26) Find the exact value of this expression sin cos 1 sin .<br />
2 2<br />
27) Write the trigonometric expression<br />
u and v.<br />
cos sin<br />
u<br />
cos<br />
1 1<br />
v as an algebraic expression containing<br />
VI. Can You Use Double-angle and Half-angle Formulas to Evaluate Trigonometric Functions<br />
and Prove Identities?<br />
A. Use Double-angle Formulas to Find Exact Values.<br />
28) Find cos(2θ) given that sin θ = 15<br />
17 , 0 < θ < π/2.<br />
29) Find cos(2θ) given that cos θ = 1<br />
3<br />
, csc θ < 0.<br />
4<br />
30) Find sin(2θ) given that sin θ =<br />
5<br />
, 3π/2 < θ < 2π.<br />
31) Find the exact value of the expression<br />
1 2<br />
sin 2sin<br />
2<br />
.<br />
B. Use Double-angle Formulas to Establish Identities.<br />
2<br />
csc<br />
32) Establish the identity sec(2 ) .<br />
2<br />
csc 2<br />
33) Establish the identity sin(4x) = (4 sin x cos x)(<br />
C. Use Half-angle Formulas to Find Exact Values.<br />
2<br />
2cos x 1).<br />
34) Find cos ( 2<br />
) given that sin θ = 1 4<br />
and tan θ > 0.<br />
35) Find sin ( 2<br />
) given that csc θ = 6 and cos θ > 0.<br />
36) Find tan ( 2<br />
) given that tan θ = 3, π < θ < 3π/2.<br />
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VII. Can You Solve Equations Involving Trigonometric Functions?<br />
A. Solve Equations Involving a Single Trigonometric Function.<br />
37) Solve the equation<br />
2<br />
4sin 1 on the interval 0 ≤ θ < 2π.<br />
38) Solve the equation 5 2 sin 4 1on the interval 0 ≤ θ < 2π.<br />
39) Solve the equation<br />
3<br />
cos(2 )<br />
2<br />
on the interval 0 ≤ θ < 2π.<br />
40) Solve the equation<br />
cos<br />
2<br />
2<br />
. Give a general formula for all the solutions.<br />
41) Use your graphing utility to solve the equation csc θ = -2 on the interval 0 ≤ θ < 2π. Round the<br />
answer to two decimal places.<br />
42) A mass hangs from a spring which oscillates up and down. <strong>The</strong> position P (in feet) of the mass at<br />
time t (in seconds) is given by P = 4cos(4 t ) . For what values of t, 0 ≤ t < π, will the position be<br />
2 2 feet? Find the exact values. Do not use a calculator.<br />
B. Solve Equations That Are Quadratic in Form.<br />
43) Solve the equation<br />
44) Solve the equation<br />
C. Solve Trigonometric Equations Using Identities.<br />
45) Solve the equation<br />
46) Solve the equation<br />
2<br />
cos 2cos 1 0 on the interval 0 ≤ θ < 2π.<br />
2<br />
2sin 3sin 2 0 on the interval 0 ≤ θ < 2π.<br />
2<br />
sin 5(cos 1) on the interval 0 ≤ θ < 2π.<br />
2<br />
cos(2 ) 6sin 2 on the interval 0 ≤ θ < 2π.<br />
47) Solve the equation sin(2θ) + sin θ = 0 on the interval 0 ≤ θ < 2π.<br />
D. Use a Calculator to Solve Equations That Include Trigonometric Functions.<br />
48) Use your graphing utility to solve the equation 6x - 5 sin x = 2. Round the solution(s) to two<br />
decimal places if necessary.<br />
2<br />
49) Use a graphing utility to solve the equation 11 24sin x 16cos xon the interval<br />
0° ≤ x < 360°. Express the solution(s) rounded to one decimal place.<br />
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<strong>MATH</strong> <strong>175</strong> <strong>Chapter</strong> 7 <strong>Review</strong>: <strong>Analytic</strong> <strong>Trigonometry</strong><br />
Answer Key<br />
1) π/3<br />
2) π/6<br />
3) -π/4<br />
4) -0.85<br />
5) 1.86<br />
6) 6π/7<br />
7) - π/6<br />
8) - π/4<br />
9)<br />
10) x =<br />
x 3<br />
f 1 x cos 1 . Domain: 4 , 10 . Range: 0 ,<br />
7<br />
3<br />
2<br />
11)<br />
2<br />
1 u<br />
12) 2 cot<br />
13)<br />
2<br />
sin x cos x cos x 2 2 2 2 2<br />
sin x(tan x cos x cot x cos x) sin x sin x cos x (1 cos x) cos x 1 2cos x.<br />
cos x<br />
sin x<br />
14)<br />
15)<br />
cosu<br />
cos u<br />
1 1<br />
cosu sin u cosu sin u 1 1 tan u<br />
1<br />
cosu<br />
1<br />
cosu<br />
sinu<br />
cosu<br />
2 2<br />
1 sin u 1 sin u 1 sin u 1 sin u 1 sin u cos u cosu<br />
secu<br />
tan u<br />
.<br />
cosu cosu cosu cosu 1 sin u cos u(1 sin u) cos u(1 sin u) 1 sin u<br />
sin x sin x sin x[1 cos x 1 cos x] 2sin x 2sin x<br />
16)<br />
2 2<br />
1 cos x 1 cos x (1 cos x)(1 cos x) 1 cos x sin x<br />
17)<br />
2 6<br />
4<br />
18) 2 3<br />
19)<br />
3<br />
3<br />
2csc x.<br />
20) 1 2<br />
21) 152<br />
377<br />
22) 304<br />
425<br />
23)<br />
3 5 1<br />
8<br />
24) cos (x + (π/6)) = cos x cos (π/6) - sin x sin (π/6) =<br />
cos x sin x.<br />
3 1<br />
2 2<br />
REV 12/01/2010 5
25)<br />
26) 0<br />
27)<br />
cos( x y) cos xcos y sin xsin y 1/(cos x cos y) cos x cos y sin xsin y 1 tan x tan y<br />
cos( x y) cos xcos y sin xsin y 1/(cos x cos y) cos x cos y sin xsin y 1 tan x tan y<br />
2 2<br />
v 1 u u 1 v<br />
28)<br />
161<br />
289<br />
29)<br />
7<br />
9<br />
30)<br />
24<br />
25<br />
31) 1<br />
32)<br />
sec(2 )<br />
1 1 csc<br />
cos(2 ) 1 2sin 2 csc 2<br />
1 2<br />
2<br />
sin<br />
2 1<br />
2<br />
sin<br />
2<br />
2<br />
33) sin(4x) = 2 sin(2x) cos(2x) = (4 sin x cos x)(2 cos x - 1).<br />
34)<br />
35)<br />
36)<br />
4<br />
15<br />
8<br />
6 30<br />
12<br />
1 10<br />
3<br />
, , ,<br />
37)<br />
5 7 11<br />
6 6 6 6<br />
38) 5 7<br />
4<br />
,<br />
4<br />
11 13 23<br />
39)<br />
12<br />
,<br />
12<br />
,<br />
12<br />
,<br />
12<br />
40) {θ| θ = (3π/4) + 2kπ, θ = (5π/4) + 2kπ} where k = 0, 1, 2,.......<br />
41) 5.76, 3.67<br />
t , , ,<br />
7 9 15<br />
42)<br />
16 16 16 16<br />
43) π<br />
44) 7 11<br />
6<br />
,<br />
6<br />
45) π<br />
5 7 11<br />
46)<br />
6<br />
,<br />
6<br />
,<br />
6<br />
,<br />
6<br />
47)<br />
2 4<br />
3 3<br />
0, , ,<br />
48) 1.06<br />
49) 48.6°, 131.4°<br />
REV 12/01/2010 6