M305G - Trigonometry Worksheet 2 - 11/6
1. Write the angles
π
4
= 450 =
1
8
and − 3π
in degrees and as a fraction of a rotation.
2
π
4
rotation counterclockwise
= −2700 =
− 3π
2
3
4
rotation clockwise
2. Give the value of all six trigonometric functions of
√
7π
.
6
√
P 7π = (− 23 , − 12 ) So, sin( 7π
) = − 21 , cos( 7π
) = −
6
6
6
7π
7π
2
csc( 6 ) = −2, and sec( 6 ) = − √3
3
,
2
tan( 7π
) =
6
√1 ,
3
) =
cot( 7π
6
√
3,
3. Solve the equation cos2 (θ) + cos(θ) = 2.
Let u = cos(θ). Then this becomes u2 + u − 2 = (u + 2)(u − 1) = 0. But cos(θ) never
equals −2, so we’re left with cos(θ) = 1, which happens for θ ∈ {2nπ}.
4. Graph the following functions using graphing transformations.
a) f (x) = sin( π2 x) − 1 One period contains (0, −1), (1, 0), (2, −1), (3, −2), and (4, −1).
, 0), and (π, −3).
b) g(x) = −3 cos(2x) One period contains (0, −3), ( π4 , 0), ( π2 , 3), ( 3π
4
c) h(x) = sec(2x) One period contains vertical asymptotes at x = − π4 , x =
and the points (0, 1) and ( π2 , −1).
x = 3π
4
π
,
4
and
12
5. Suppose that sin(θ) = 13
and that cos(θ) < 0. What quadrant is θ in? Use this to find
all six trigonometric functions of θ. Finally, use this, and the half angle formula, to find
an exact value for cos( 2θ ).
12
5
13
Second quadrant: sin(θ) = 13
, cos(θ) = − 13
, tan(θ) = − 12
, csc(θ) = 12
, sec(θ) = − 13
,
5
5
5
cot(θ) = − 12
q 5
q
1− 13
1+cos(θ)
θ
cos( 2 ) = ±
=±
= ± √213 , but 2θ is in the first quadrant, so keep positive
2
2
answer.
6. Find exact values for the following expressions. Explain, don’t just write a number.
a) sin−1 (sin(60 )) = 60
b) sin−1 (sin(3660 )) = 60
c) sin−1 (sin(1740 )) = 60
d) sec−1 (.8) is undefined
7. Verify the following identities.
a) L = cos(θ) cot(θ) sec(θ) tan(θ) = 1
1 sin(θ)
=1
L = cos(θ) cos(θ)
sin(θ) cos(θ) cos(θ)
b) L =
L=
sin2 (θ)−cos2 (θ)
= sin(θ) − cos(θ)
sin(θ)+cos(θ)
(sin(θ)+cos(θ))(sin(θ)−cos(θ))
= sin(θ)
sin(θ+cos(θ)
c) L = 1 −
L=
cos2 (θ)
1+sin(θ)
1+sin(θ)
1+sin(θ)
−
− cos(θ)
= sin(θ)
cos2 (θ)
1+sin(θ)
=
1+sin(θ)−(1−sin2 (θ))
1+sin(θ)
=
sin(θ)+sin2 (θ)
1+sin(θ)
=
sin(θ)(1+sin(θ))
1+sin(θ)
= sin(θ)
d) L = ln | sec(θ) + tan(θ)| + ln | sec(θ) − tan(θ)| = 0
L = ln |(sec(θ) − tan(θ))(sec(θ) + tan(θ))| = ln |sec2 (θ) − tan2 (θ)| = ln |1| = 0
8. Find exact values for the following expressions. Show your work or write a sentence for
full credit.
a) cot(cot−1 (2)) = 2
c)
d)
e)
√
2
2
5
sin(cot−1 ( 12
)) = 13
5
sec(cos−1 (x)) = x1
sin(2 cos−1 ( 35 )) = 2 · 45
b) cos(tan−1 (1)) =
·
3
5
9. Use the sum and difference formulae for sine and cosine to find exact values for the
following expressions. Tell me which formula you are using and label the angles. This
means that if there is an α in the formula, tell me what it is in your case.
π
a) sin( 12
) = sin( π3 − π4 ) = sin( π3 ) cos( π4 ) − cos( π3 ) sin( π4 ) =
√ √
3 2
2 2
−
√
1 2
2 2
0
b) sin(150 ) cos(450 ) − cos(150 ) sin(450 ) = sin(150 − 450 ) = sin(−30 ) = − 12
c) Show that tan( π2 − θ) = cot(θ). tan( π2 − θ) =
1·cos(θ)−0·sin(θ)
0·cos(θ)+1·sin(θ)
=
cos(θ)
sin(θ)
= cot(θ) (as in the book)
sin( π2 −θ)
cos( π2 −θ)
A sine without an angle is a SIN.
=
sin( π2 ) cos(θ)−cos( π2 ) sin(θ)
cos( π2 ) cos(θ)+sin( π2 ) sin(θ)
=