Math 1720 - Trigonometry
University of Memphis
Instructor: Gábor Mészáros
Central European University, Budapest
10.28.2013
Solutions
I.
1. Solve the following trigonometric equations on
(a)
(b)
sec4 x − 4 sec2 x + 4 = 0,
4
Introduce
[0, 2π]!
z = sec2 x
2
and solve the quadratic equation on z...
2
cos x”2 sin x − 1 = 0, Use the identity sin x + cos2 x = 1
z = cos2 x and solve the quadratic equation on z...
to get rid of
sin2 x.
Introduce
2. Prove the following identities:
(a)
sec α sin α = tan α,
(b)
1 + cot2 (−α) = csc2 α,
Substitute
cot2 (−α) = cot2 (α) =
(c)
cos2 α(1 + tan2 α) = 1.
cos2 α
.
sin2 α
sec x =
Here
1+
1
sin x .
cos2 α
sin2 α
=
cos2 α+sin2 α
sin2 α
=
1
sin2 α
= csc2 α.
Use the same technique as above.
II.
Recall the following trigonometric identities:
sin(α + β) = sin α cos β + sin β cos α
(1)
sin(α − β) = sin α cos β − sin β cos α
(2)
sin(2α) = 2 sin α cos α
(3)
1. Determine the exact value of the following functions at the given numbers:
(a)
π
),
sin( 12
√
√
2( 3−1)
.
4
√ √
sin π6 cos π4 = 2( 43+1) .
√ √
sin π3 cos π4 = 2( 43+1) .
7π
5π
12 ) = sin( 12 )
π
sin( 12
) = sin( π4 − π6 ) = sin π4 cos π6 − sin π6 cos π4 =
(b)
5π
π
π
π
π
sin( 5π
12 ), sin( 12 ) = sin( 4 + 6 ) = sin 4 cos 6 +
(c)
5π
π
π
π
π
sin( 7π
12 ), sin( 12 ) = sin( 4 + 3 ) = sin 4 cos 3 +
Or one can simply say that
(d)
sin( 7π
12 ) = sin(π −
sin 80◦ cos 20◦ − sin 20◦ cos 80◦ .
Observe that we are calculating
sin(80◦ − 20◦ ) = sin(60◦ ) =
1
√
3
2 .
2. Solve the trigonometric equation
Use identity (1), that gives
sin α cos α = − 21 .
sin(2α) = −1. α =
2
3π
4
+ nπ .