Download Trigonometric Identities - The University of Akron

THE UNIVERSITY OF AKRON
Theoretical and Applied Mathematics
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c 2003 [email protected]
Last Revision Date: April 12, 2003
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Katie Jones
and
Tom Price
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Flash Cards
Trigonometric Identities
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tan2 +1 = sec2 t.
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Verify that
Hint
Soln
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2 sin t + cos2 t = 2.
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Solve the equation
Hint
Soln
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Show that
π
− α = tan α.
cot
2
Hint
Soln
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AcroTEX eDucation Bundle
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Without a calculator determine
the value of
7π
sin .
12
Hint
Soln
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cos 285◦.
WebTrig Flash Cards
Without a calculator determine
the value of
Hint
Soln
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tan 75◦.
WebTrig Flash Cards
Without a calculator determine
the value of
Hint
Soln
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Without a calculator determine
the value of
5π
tan .
12
Hint
Soln
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Without a calculator determine
the value of
13π
sin
.
12
Hint
Soln
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cos(2α) = 2 cos2 α − 1.
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Verify the double-angle formula
Hint
Soln
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Verify the half-angle formula for
the sine function:
α
1 − cos α
sin = ±
.
2
2
Hint
Soln
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AcroTEX eDucation Bundle
sin 15◦.
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Without a calculator determine
the value of
Hint
Soln
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Without a calculator determine
the value of
8π
cos .
3
Hint
Soln
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Verify the cofunction identity
π
− α = cos α
sin
2
π
for the value of sin − 6 .
Hint
Soln
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cos α sin β
1
= [sin (α + β) − sin (α − β)]
2
for the values
π
5π
α = and β = .
3
6
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Verify the product formula
Hint
Soln
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α+β
α−β
= 2 cos
cos
2
2
for the values
π
2π
α = and β = .
3
3
AcroTEX eDucation Bundle
cos α + cos β
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Verify the factor formula
Hint
Soln
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Simplify the expression
cos2 t
.
1 − sin t
Hint
Soln
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sec t cot t = csc t.
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Show that
Hint
Soln
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sin t cot t = cos t.
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Show that
Hint
Soln
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tan (−t) = − tan t.
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Verify that
Hint
Soln
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cos 3a = 4 cos3 a − 3 cos a.
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Show that
Hint
Soln
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HINT
use the Pythagorean identity involving sine
and cosine.
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tan2 +1 = sec2 t
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To verify that
Hint
Soln
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Solution: Divide both sides of the Pythagorean identity
sin2 t + cos2 t = 1
AcroTEX eDucation Bundle
=⇒
1
sin2 t
+1=
cos2 t
cos2 t
tan2 +1 = sec2 t.
WebTrig Flash Cards
by cos2 t to obtain
Hint
Soln
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HINT
the pythagorean identity
sin2 t + cos2 t = 1.
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2 sin t + cos2 t = 2
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To solve the equation
Hint
Soln
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Answer: t =
π
2
+ 2πk
=⇒
2 sin t + 1 − sin2 t = 2
=⇒
2 sin t − 1 − sin2 t = 0
=⇒
sin2 t − 2 sin t + 1 = 0
=⇒
(sin t − 1) = 0
sin t = 1
=⇒
2
The sine function equals 1 whenever t =
integer k.
π
2
AcroTEX eDucation Bundle
2 sin t + cos2 t = 2
WebTrig Flash Cards
Solution: We know that sin2 t + cos2 t = 1, so cos2 t =
1 − sin2 t. Hence,
+ 2πk for some
Hint
Soln
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To show that π
− α = tan α
cot
2
use the cofunction identity involving tangent.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Solution: Using the definition of the cotangent function
and a similar cofunction identity for the tangent function, we
have
π
1
1
cot( − α) =
=
= tan α.
2
tan( π2 − α)
cot α
Hint
Soln
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HINT
=
π
3
+ π4 .
AcroTEX eDucation Bundle
7π
12
WebTrig Flash Cards
To find sin 7π
, notice that
12
Hint
Soln
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Answer: sin
7π
12
√
=
√
6+ 2
4
=
=
√
√
√
3
2
2
2 · 2 + 2
√
√
6
2
4 + 4
√ √
6+ 2
.
4
·
1
2
AcroTEX eDucation Bundle
=
WebTrig Flash Cards
π
π
Solution: Notice 7π
12 = 3 + 4 , which are both basic angles
we know from the unit circle. So,
π π
sin 7π
12 = sin 3 + 4
= sin π3 cos π4 + sin π4 cos π3
Hint
Soln
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HINT
cos (a + b) = cos a cos b − sin a sin b.
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cos 285◦
use this the sum formula for the cosine function
WebTrig Flash Cards
To find
Hint
Soln
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◦
Answer: cos 285 =
√ √
− 2− 6
4
= cos 240 cos 45 − sin 240 sin 45
√ √
√
= − 12 · 22 + − 23 · 22
√ √ = − 42 + − 46
=
√ √
− 2− 6
.
4
AcroTEX eDucation Bundle
cos 285◦ = cos (240◦ + 45◦ )
WebTrig Flash Cards
Solution: Notice 285 = 240 + 45. So,
Hint
Soln
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HINT
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tan 75◦
a the difference formula for tangent.
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To find
Hint
Soln
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◦
Answer: tan 75 =
√
−1−√ 3
1− 3
AcroTEX eDucation Bundle
tan 75◦ = tan (135◦ − 60◦ )
tan 135 − tan 60
=
1 + tan 135 tan 60
√
−1 − 3
√
=
1 + (−1) 3
√
−1 − 3
√ .
=
1− 3
WebTrig Flash Cards
Solution: Notice 75 = 135 − 60. So,
Hint
Soln
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HINT
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5π
12
find two numbers whose sum or difference
and use the sum formula for tangent
is 5π
12
function.
tan
WebTrig Flash Cards
To find
Hint
Soln
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Answer: tan
5π
12
=
√
1−
√ 3
3−1
WebTrig Flash Cards
AcroTEX eDucation Bundle
5π
π
π
12 = 6 + 4 . So,
π π
tan 5π
12 = tan 6 + 4
tan π6 + tan π4
=
1 − tan π6 tan π4
√1 − 1
3
=
1 − √13 · 1
Solution: Notice
√
1− 3
=√
.
3−1
Hint
Soln
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HINT
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13π
12
use the difference formula for the sine function: sin (a − b) = sin a cos b − sin b cos a.
sin
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To find
Hint
Soln
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Answer: sin
13π
12
π
= 4π
3 − 4 . So,
4π π = sin 3 − 4
cos π − sin π cos 4π
= sin 4π
√3 √ 4 √ 4 3
= − 23 22 − 22 − 12
=
=
√
√
− 6
2
+
4
4
√ √
− 6+ 2
.
4
AcroTEX eDucation Bundle
sin 13π
12
=
√ √
− 6+ 2
4
WebTrig Flash Cards
Solution: Notice
13π
12
Hint
Soln
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HINT
use the fact that 2a = a + a.
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cos(2α) = 2 cos2 α − 1
WebTrig Flash Cards
To verify the double-angle formula
Hint
Soln
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Solution: Since 2a = a + a, use the cosine formula for the
sum of two angles and a pythagorean identity to solve
= cos2 α − (1 − cos2 α)
= 2 cos2 α − 1.
AcroTEX eDucation Bundle
= cos2 α − sin2 α
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cos(2α) = cos(α + α)
= cos α cos α − sin α sin α
Hint
Soln
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HINT
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cos(2t) = 1 − 2 sin2 t.
WebTrig Flash Cards
To verify the half-angle formula for the sine
function:
1 − cos α
α
,
sin = ±
2
2
start with the double angle identity
Hint
Soln
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Solution: By the double angle formula for the cosine function we have
2 sin2 t = 1 − cos(2t)
Dividing this last equation through by 2 and then taking the
square root of both sides yields
1 − cos(2t)
.
sin t = ±
2
Replacing t with α2 in this equation yields the desired halfangle formula
1 − cos α
α
.
sin = ±
2
2
AcroTEX eDucation Bundle
=⇒
WebTrig Flash Cards
cos(2t) = 1 − 2 sin2 t
Hint
Soln
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find sin 15◦ , notice that
30
15 = .
2
Hint
Soln
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Answer: sin
√
=
√
2− 3
2
AcroTEX eDucation Bundle
◦
Solution: Notice 15 = 30
2 , and 30 is a known basic angle.
So, use the half angle formula for the sine function to solve.
√
√
√
3
◦
1− 2
1 − cos 30
2− 3
2− 3
=
=
=
.
=
2
2
4
2
WebTrig Flash Cards
30 ◦
2
Hint
Soln
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To find cos 8π
, notice that
3
4π
8π
=2·
3
3
and use the double angle formula for cosine.
Hint
Soln
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1
Answer: cos 8π
3 = −2
AcroTEX eDucation Bundle
WebTrig Flash Cards
4π
Solution: Notice 8π
3 = 2 · 3 and use the double angle
formula for cosine to obtain
cos 2 · 4π
= 2 cos2 4π
3
3 −1
2
1
=2 −
−1
2
1
−1
=2
4
1
=− .
2
Hint
Soln
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To verify the cofunction identity
π
− α = cos α
sin
2
for the value of sin − π6 , first set up an
equation to find α.
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Solution: First find α by solving
π
π
− = −α
6
2
π π
2π
=⇒
α= + =
.
2
6
3
Now, evaluate
= − 12 = cos 2π
sin − π6 = sin π2 − 2π
3
3 .
Hint
Soln
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HINT
AcroTEX eDucation Bundle
WebTrig Flash Cards
To verify the product formula
1
cos α sin β = [sin (α + β) − sin (α − β)]
2
for the values
5π
π
.
α = and β =
3
6
use
.
sin (α + β) = sin π3 + 5π
6
Hint
Soln
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Answer: Both yield the value
1
4
Also,
1
[sin (α + β) − sin (α − β)] =
2
=
1
2
·
1
2
= 14 .
π 5π sin 3 + 6 − sin π3 −
7π π 1
2 sin 6 − sin − 2
= 12 − 12 − (−1) = 14 .
1
2
5π
6
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cos α sin β = cos π3 sin 5π
6 =
WebTrig Flash Cards
Solution: In this case,
Hint
Soln
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HINT
= 2 cos
α+β
2
for the values α =
= − π6 .
that α−β
2
π
3
cos
and β =
α−β
2
2π
,
3
observe
AcroTEX eDucation Bundle
cos α + cos β
WebTrig Flash Cards
To verify the factor formula
Hint
Soln
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Answer: Both yield the value 0
WebTrig Flash Cards
Solution: In this case,
Also,
π 2π π 2π + 3
− 3
α+β
α−β
2 cos
cos
= 2 cos 3
cos 3
2
2
2
2
π
−π = 2 cos 2 cos 6
√ = 2 (0) 23 = 0.
AcroTEX eDucation Bundle
2π
1
1
π
= + −
= 0.
cos α + cos β = cos + cos
3
3
2
2
This means that both sides of the given equation are zero
which establishes the desired equality.
Hint
Soln
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HINT
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cos2 t = 1 − sin2 t.
WebTrig Flash Cards
To simplify the expression
cos2 t
1 − sin t
use the pythagorean identity
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Solution: Use the Pythagorean identities and factoring to
establish
cos2 t
1 − sin2 t
=
1 − sin t
1 − sin t
(1 − sin t) (1 + sin t)
=⇒
=
1 − sin t
=⇒
= 1 + sin t.
Hint
Soln
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HINT
use the basic reciprocal identities
1
cos t
.
sec t =
and cot t =
cos t
sin t
AcroTEX eDucation Bundle
sec t cot t = csc t
WebTrig Flash Cards
To show that
Hint
Soln
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AcroTEX eDucation Bundle
WebTrig Flash Cards
Solution: Use the reciprocal identities to solve.
cos t
1
·
sec t cot t =
cos t sin t
1
=
sin t
= csc t.
Hint
Soln
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HINT
use the basic reciprocal identities.
AcroTEX eDucation Bundle
sin t cot t = cos t
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To show that
Hint
Soln
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WebTrig Flash Cards
Solution: Use the reciprocal identities to solve.
cos t
sin t cot t = sin t ·
sin t
= cos t.
Hint
Soln
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HINT
remember that the sine function is odd and
cosine is even. That is,
sin (−t) = − sin t and cos (−t) = cos t.
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tan (−t) = − tan t,
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To verify that
Hint
Soln
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Solution: Use the symmetry identities to solve, remembering that the sine function is odd and cosine is even.
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sin (−t)
cos (−t)
− sin t
=
cos t
= − tan t.
tan (−t) =
AcroTEX eDucation Bundle
Hint
Soln
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HINT
use the fact that 3a = 2a + a.
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cos 3a = 4 cos3 a − 3 cos a
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To show that
Hint
Soln
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= 2 cos3 a − cos a − 2 cos a sin2 a
= 2 cos3 a − cos a − 2 cos a 1 − cos2 a
= 2 cos3 a − cos a − 2 cos a + 2 cos3 a
= 4 cos3 a − 3 cos a.
AcroTEX eDucation Bundle
cos 3a = cos (2a + a)
= cos 2a cos a − sin 2a sin a
= 2 cos2 a − 1 cos a − 2 sin a cos a sin a
WebTrig Flash Cards
Solution: Use the fact that 3a = 2a + a and the sum,
difference, Pythagorean, and double angle formulas to arrive
at the following:
Hint
Soln
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