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TRIGONOMETRY
the other
inverse trig functions
math
hands
CH 02 SEC 06 SOLUTIONS
For each angle, draw and label a reference triangle, then determine all 6 trig ratios, sine , cosine, tangent, secant, cosecant,
and cotangent. Do not use calculators here.
1. Find the angle θ
Solution:
6
5
In this case, we know the opposite and the hypothenuse sides. The function describing such ratio is the sine,
thus
5
sin θ =
6
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
5
θ = sin−1
6
≈ 0.985
θ
(in radians [see calculator mode]... OR...)
◦
≈ 56.443
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
sin θ = 65 under a different context, where we will solve it
completely, not limited to the domain and codomain of the
arcsin function.
2. Find the angle θ
Solution:
3
θ
2
In this case, we know the adjacent and the hypothenuse sides. The function describing such ratio is the
cosine, thus
2
cos θ =
3
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
2
θ = cos−1
3
≈ 0.841
≈ 48.19◦
(in radians [see calculator mode]... OR...)
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
cos θ = 32 under a different context, where we will solve it
completely, not limited to the domain and codomain of the
arccos function.
pg. 1
c
2007-2011
MathHands.com v.1012
TRIGONOMETRY
the other
inverse trig functions
math
hands
CH 02 SEC 06 SOLUTIONS
3. Find the angle θ
Solution:
11
2
In this case, we know the opposite and the hypothenuse sides. The function describing such ratio is the sine,
thus
2
sin θ =
11
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
2
θ = sin−1
11
≈ 0.183
θ
≈ 10.476◦
(in radians [see calculator mode]... OR...)
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
2
under a different context, where we will solve it
sin θ = 11
completely, not limited to the domain and codomain of the
arcsin function.
4. Find the angle θ
Solution:
5
θ
2
In this case, we know the opposite and the adjacent sides. The function describing such ratio is the tangent,
thus
5
tan θ =
2
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
5
θ = tan−1
2
≈ 1.19
≈ 68.199◦
(in radians [see calculator mode]... OR...)
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
tan θ = 25 under a different context, where we will solve it
completely, not limited to the domain and codomain of the
arctan function.
pg. 2
c
2007-2011
MathHands.com v.1012
TRIGONOMETRY
the other
inverse trig functions
math
hands
CH 02 SEC 06 SOLUTIONS
5. Find the angle θ
Solution:
5
θ
1
In this case, we know the opposite and the adjacent sides. The function describing such ratio is the tangent,
thus
5
tan θ =
1
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
5
θ = tan−1
1
≈ 1.373
≈ 78.69◦
(in radians [see calculator mode]... OR...)
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
tan θ = 15 under a different context, where we will solve it
completely, not limited to the domain and codomain of the
arctan function.
6. Find the angle θ
Solution:
4
θ
3
In this case, we know the adjacent and the hypothenuse sides. The function describing such ratio is the
cosine, thus
3
cos θ =
4
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
3
θ = cos−1
4
≈ 0.723
◦
≈ 41.41
(in radians [see calculator mode]... OR...)
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
cos θ = 43 under a different context, where we will solve it
completely, not limited to the domain and codomain of the
arccos function.
pg. 3
c
2007-2011
MathHands.com v.1012
TRIGONOMETRY
CH 02 SEC 06 SOLUTIONS
the other
inverse trig functions
math
hands
7. Find the angle θ
Solution:
9
θ
2
In this case, we know the adjacent and the hypothenuse sides. The function describing such ratio is the
cosine, thus
2
cos θ =
9
Since this is not a famous ratio, we allow ourselves use of a
calculator to estimate the sought angle.
2
θ = cos−1
9
≈ 1.347
≈ 77.16◦
(in radians [see calculator mode]... OR...)
(in degrees [see calculator mode])
... it should be noted that we will revisit the equation
cos θ = 92 under a different context, where we will solve it
completely, not limited to the domain and codomain of the
arccos function.
8.
cos−1 (cos(30◦ )) = 30◦
A. TRUE
B. FALSE
9.
cos−1 (cos(−30◦ )) = −30◦
A. TRUE
B. FALSE
10.
tan−1 (tan(30◦ )) = 30◦
A. TRUE
B. FALSE
11.
tan−1 (tan(210◦ )) = 210◦
A. TRUE
pg. 4
B. FALSE
c
2007-2011
MathHands.com v.1012
TRIGONOMETRY
CH 02 SEC 06 SOLUTIONS
12.
math
hands
tan tan
A. TRUE
−1
the other
inverse trig functions
1
1
=
3
3
B. FALSE
13.
5
5
tan tan−1
=
3
3
A. TRUE
B. FALSE
14.
5
5
=
cos cos−1
13
13
A. TRUE
B. FALSE
15. Compute
sin cos−1 (.3456)
Solution: First, the stuff inside, cos−1 (.3456) ≈ 69.78◦ then...
sin (69.78◦) ≈ 0.9384
16. Compute
tan cos−1 (.1234)
Solution: First, the stuff inside, cos−1 (.1234) ≈ 82.91◦ then...
tan (82.91◦) ≈ 8.0418
note: the self quiz on ”mission accomplished” prob #2 outlines a different approach to this same question.
pg. 5
c
2007-2011
MathHands.com v.1012