Download Math 110 Some answers to sample test 2 1. Do p. 759 #32. csc( t

Math 110 Some answers to sample test 2
1. Do p. 759 #32.
csc( t ) = 5 with π2 < t < π. Find the exact values of the other trigonometric functions.
csc( t ) = 5 = 5 / 1.
√
2
By the Pythagorean Theorem x√
+ 12 = 52 or x2 + 1 = 25 or x2 = 25 − 1 = 24 or x = ± 24
Since t is in quadrant 2, x = − 24.
√
So the associated right triangle has a horizontal leg of − 24 on the negative x axis,
a vertical leg of 1 and a hypotenuse of 5.
sin( t ) = 1 / csc( t ) = 1/5. √
.
cos( t ) = x / hypotenuse = − 5 24 . sec( t ) = 1 / cos( t ) = √−5
24
√
−1
tan( t ) = − 24. cot( t ) = √24 .
2. Do p. 760 #60.
Solve sec( t ) = - 2 / sqrt( 3 ).
Invert to obtain cos( t ) = -sqrt( 3 ) / 2.
cosine is negative in quadrants 2 and 3. The reference angle is π/6.
So t = 5π/6 in quadrant 2 and t = 7π/6 in quadrant 4.
Thus all solutions are t = 5π/6 + 2kπ or 7π/6 + 2kπ where k is an integer.
3. Do p. 762 #80.
From the observation deck of the lighthouse at Sasquatch Point 50 feet above the surface of
Lake Ippizuti,a lifeguard spots a boat out on the lake sailing directly toward the lighthouse.
The first sighting had an angle of depression of 8.2 degrees and the second sighting had
an angle of 25.9 degrees. How far had the boat traveled between the sightings?
With reference to the figure, find PQ.
Here TB = 50 feet. Angle TPB is 8.2 degrees. Angle TQB is 25.9 degrees.
Find PQ as PB - QB.
tan( TQB ) = TB / QB. So QB = TB / tan( TQB ) = 50 / tan( 25.9◦ )
tan( TPB ) = TB / PB. So PB = TB / tan( TPB ) = 50 / tan( 8.2◦ ).
So PQ = PB - QB = 50 / tan( 25.9◦ ) - 50 / tan( 8.2◦ ) = 146.4 feet.
4. Do p. 762 #91. Show
cos(t)
Left = 1−sin(t)
2
cos(t)
1
= cos(t)2 = cos(t)
cos(t)
1−sin(t)2
= sec(t).
Use a Pythogoran identity on the bottom.
= sec(t)
1
1
5. Do p. 763 #116. Show csc(t)−cot(t)
− csc(t)+cot(t)
= 2cot(t).
1
1
B−A
Use the algebraic identity A − B = AB on the left side.
csc(t)+cot(t)−(csc(t)−cot(t))
1
1
Left = csc(t)−cot(t)
− csc(t)+cot(t)
= (csc(t)+cot(t))·(csc(t)−cot(t))
Simplify the top. FOIL the bottom.
=
=
=
2cot(t)
csc(t)2 −csc(t)cot(t)+cot(t)csc(t)−cot(t)2
2cot(t)
Use a Pythagorean
csc(t)2 −cot(t)2
2cot(t)
= 2cot(t) = right
1
Simplify the bottom.
identity. 1 + cot(t)2 = csc(t)2
6. State the domain and range of f( t ) = tan( t ).
The domain of tan( t ) is all real numbers except where cos( t ) = 0 since tan( t ) = sin( t ) / cos( t ).
Now cos( t ) = 0 for t = π2 + kπ for an integer k.
1
So the domain of tan( t ) is all real numbers t except t =
The range of tan( t ) is all real numbers.
π
2
+ kπ for an integer k.
7. Do p. 763 #125.
1
Show 1+cos(t)
= csc(t)2 − csc(t)cot(t)
Start with the right side. Trade in for sine and cosine.
cos(t)
1−cos(t)
1
1
Use sin(t)2 = 1 − cos(t)2 = (1 − cos(t)) · (1 + cos(t))
Right = sin(t)
2 − sin(t) · sin(t) = sin(t)2
Right =
1−cos(t)
(1−cos(t))(1+cos(t))
=
1
1+cos(t)
8. Do p. 782 #20. Find the exact value of csc( 5 Pi / 12 ).
Notice 5/12 = 2/3 - 1/4. Use the sine of a difference formula.
1
1
1
csc( 5π
12 ) = sin( 5π ) = sin( 2π − π ) = sin( 2π )cos( π )−cos( 2π )sin( π ) .
12
=
1
−1 √
3 √
1
1
2 · 2− 2 · 2
√
3
=
1
√
3+1
√
2 2
=
4√
2 2
√
3+1
3
4
3
4
9. Do p. 782 #22.If A is a Quadrant IV angle with cos( A ) = √15 and sin( B ) = √110
where π/2 < B < π, then find a) cos( A + B ) b) sin( A + B ) c) tan( A + B )
d) cos( A - B ) e) sin( A - B ) f) tan( A - B ).
−2
and cos( B ) = √−3
By the Pythagoran Theorem sin( A ) = √
5
10
To find parts a b and c use sum formulas. To find parts d e and f use difference formulas.
−2 √1
√
−√
· 10 = 5−1
cos( A + B ) = cos( A ) cos( B ) - sin( A ) sin( B ) = √15 · √−3
10
5
2
−2
−3
1
1
7
sin( A + B ) = sin( A ) cos( B ) + cos( A ) sin( B ) = √5 · √10 + √5 · √10 = 5√
2
tan( A + B ) = sin( A + B )/cos( A + B ) =
7
√
5 2
−1
√
5 2
= −7 from the previous 2 lines.
Now use the difference formulas.
−2 √1
√ =
cos( A - B ) = cos( A ) cos( B ) + sin( A ) sin( B ) = √15 · √−3
+√
· 10 = 5−5
10
5
2
−2 √
−3
1 √1
5
√
√
√
sin( A - B ) = sin( A ) cos( B ) - cos( A ) sin( B ) = 5 10 − 5 10 = 5 2 = √12
tan( A - B ) = sin( A - B )/cos( A - B ) = -1 from the previous 2 lines.
10. Do p. 783 #31.
Show cos( A + B ) + cos( A - B ) = 2 cos( A ) cos( B )
Use the sum and difference formulas for cosine.
Left = cos( A ) cos( B ) - sin( A ) sin( B ) + cos( A ) cos( B ) + sin( A ) sin( B )
= 2 cos( A ) cos( B ).
2
−1
√
2