Download Unit 8: Trigonometry – Part 2 Revisiting Tangent, Cotangent, Secant

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AMHS Precalculus - Unit 8
Unit 8: Trigonometry – Part 2
Revisiting Tangent, Cotangent, Secant and Cosecant
These are called the Quotient Identities:
a)
tan(α ) =
sin(α )
cos(α )
b) cot(α ) =
cos(α )
sin(α )
The following are called the Reciprocal Identities:
1
sin(α )
1
c) cot(α ) =
tan(α )
1
e) sin(α ) =
csc(α )
a)
csc(α ) =
1
cos(α )
1
d) tan(α ) =
cot(α )
1
f) cos(α ) =
sec(α )
b) sec(α ) =
Ex. 1: Evaluate all six trigonometric functions at the following values of θ :
a) θ = −
π
6
b) θ =
π
2
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AMHS Precalculus - Unit 8
The Pythagorean Identities:
a) sin 2 ( x) + cos 2 ( x) =
1
Using the Quotient and Reciprocal identities we can derive the other two Pythagorean Identities:
sin 2 ( x) + cos 2 ( x) =
1
sin 2 ( x) + cos 2 ( x) =
1
Conclusion – The other two Pythagorean Identities are:
b)
c)
Ex. 2: Find the values of all six trigonometric functions from the given information:
a) sin(θ ) =
4
, θ is in the first quadrant.
5
b) csc(α ) = −5 ,
3π
< α < 2π
2
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AMHS Precalculus - Unit 8
Graphs of the Tangent and Cotangent Functions
Ex. 1: Graph f ( x) = tan x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
Ex. 2: Graph f ( x) = cot x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
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AMHS Precalculus - Unit 8
Ex.3: Sketch one period of each function
a)
f ( x) = 2 tan( x)
b)
π
f=
( x) cot( x + )
4
Ex. 4: Find the period of the following functions:
f ( x) = tan(2 x)
γ
f (γ ) = tan( )
2
Ex. 5: Find all the values of t in the interval [0, 2π ] satisfying the given equation:
a)
tan(t ) + 1 =
0
b) cot(t ) + 3 =
0
f ( x) = cot(
πx
3
)
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AMHS Precalculus - Unit 8
Graphs of the Secant and Cosecant Functions
Ex. 1: Graph f ( x) = sec x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
Ex. 2: Graph f ( x) = csc x
Domain:
Range:
x -intercepts:
Period:
Even or odd?
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AMHS Precalculus - Unit 8
Ex.3: Graph one period of each function
a) =
f ( x)
1
π
sec( x − )
2
4
b) =
f ( x) 2 csc( x −
π
2
)
More on Trigonometric Identities
Ex. 1: Use the identities you have learned so far to verify the following:
a) sin(θ ) cos 2 (θ ) − sin(θ ) =
− sin 3 (θ )
b) (1 − (cos x) 2 )(sec x) 2 =
(tan x) 2
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AMHS Precalculus - Unit 8
c) 1 + 2sin α cos α =
(sin α + cos α ) 2
d) cot x + tan x =
sec x csc x
Sum and Difference formulas for Sine and Cosine
sin(α=
+ β ) sin α cos β + cos α sin β
sin(α=
− β ) sin α cos β − cos α sin β
cos(α
=
+ β ) cos α cos β − sin α sin β
cos(α
=
− β ) cos α cos β + sin α sin β
Ex. 2: Use the sum and difference formulas to determine the value of the following trigonometric
functions.
a) sin(
π
6
b) cos(
+
3π
)
4
7π
)
12
Ex. 3: Verify the identity:
sin(t +
π
2
)=
cos t
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AMHS Precalculus - Unit 8
Double Angle Formulas
We use the sum and difference formulas to derive the double angle formulas.
1.
sin(2 x) = 2sin( x) cos( x)
2.
cos(2
=
x) cos 2 ( x) − sin 2 ( x)
Verification:
We can then use the Pythagorean identities to derive two other versions of the double angle formula for
cosine.
cos(2 x) = 1 − 2sin 2 ( x)
4. =
cos(2 x) 2 cos 2 ( x) − 1
3.
Verification:
Power-Reducing formulas
If we solve for sin 2 ( x) and cos 2 ( x) in #3 and #4 above, we get:
a)
cos 2 ( x) =
cos(2 x) + 1
2
b) sin 2 ( x) =
1 − cos(2 x)
2
These formulas should be memorized and are very useful in integral calculus.
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AMHS Precalculus - Unit 8
Ex. 4: If sin t =
−3
3π
, π <t <
¸ find cos(2t ),sin(2t ) and tan(2t )
5
2
Ex. 5: Verify the identities
=
a) cos x sin x sin 2 x + cos x cos 2 x
2 cot 2 x
b) cot x − tan x =
Ex. 6: Find all the values of x in the interval [0, 2π ] that satisfy the given equation.
a) sin 2 x = sin x
b) (cos x) 2 − 3sin x − 3 =
0
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AMHS Precalculus - Unit 8
Ex. 7: Find ALL values of t that satisfy the given equation.
a)
c)
2 cos t = − 2
(cos t ) 2 + cos t =
0
e) sin(3t ) =
− 3
2
g) 3 tan(2t ) + 3 =
0
b) sin t =
1
2
d) 2sin 2 t − sin t − 1 =
0
f)
csc t = −2
h) sin t = cos t