Download 5.1 Fundamental Identities

Math 150: §5.1 Fundamental Identities
Identities Warm-Up
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2
1. Simplify: a b =
1 1
+
a b
2. Solve sin2 x + cos2 x = 1 for sin x , and write all six trig ratios in terms of cos x .
(
3. Show that 1 − cos x
4. Add and simplify:
)
2
is equivalent to 2 − 2cos x − sin2 x .
sin x
1 − cos x
+
1 − cos x
sin x
5. Evaluate without a calculator:
a) sin2 α + cos2 α , ∀α ∈
b) sin2
2π
17
+ cos2
2π
17
( )
( )
()
()
c) csc2 − 21 − cot2 − 21
e) tan2 50o − sec2 50o
d) cot2 π − csc2 π
f) sec2 90o − tan2 90o
6. Prove the following identities algebraically (use a LS = RS QED “proper” proof
format) and verify graphically.
a) sin3 x + sin x cos2 x = sin x
© Raelene Dufresne 2011
b)
tan2 θ + 1
1 + cot θ
2
= tan2 θ
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Math 150: §5.1 Fundamental Identities
Fundamental Trig Identities:
State the following trigonometric identities and state their domains of validity.
1. Reciprocal Identities (6)
2. Quotient Identities (2)
3. Pythagorean Identities (3)
a) Use your calculator to evaluate sin2 x + cos2 x for
x = 30o
x=
π
2
x = 149o
b) Conjecture the identity value for sin2 x + cos2 x :
c) State the domains of
y = cos x y = sin x y = sin x + cos x y = sin2 x
y = cos2 x
x=−
5π
11
y = sin2 x + cos2 x
d) Prove that sin2 x + cos2 x = 1 is an identity and state its domain of validity.
e) Why is this identity called a Pythagorean Identity?
f) Divide the equation sin2 x + cos2 x = 1 by sin2 x and simplify using the quotient
and reciprocal identities. You have just DERIVED a second Pythagorean Identity.
What is its domain of validity?
g) Derive the remaining Pythagorean Identity, and state its domain of validity.
© Raelene Dufresne 2011
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Math 150: §5.1 Fundamental Identities
4. Cofunction Identities (6)
a) How are complementary angles related?
b) Determine the complements of the following angles:
(i) 30o
(ii) 89o
(iii)
π
4
(iv)
π
3
(v)
2π
15
c) How are the two acute angles of any right triangle related?
d) Sketch a right triangle with a hypotenuse of side c and leg sides of a and b; draw
an angle of θ opposite side a. What name can be given to the remaining acute
angle without introducing a new variable? (Use radians.) Label this angle.
e) Use your triangle to write all possible trig functions (you may use either angle in
the ratio) that yield each of the following ratios. The first one has been done.
Ratio
Ratio
Ratio
a
a
c
b
a
b
b
c
c
a
b
sin θ
c
f) Complete the following sentences:
(i) the sine of an angle is the ___________ of the angle’s complement.
(ii) the tangent of an angle is the ___________ of the complement.
(iii) the secant of an angle is the ___________ of the complement.
(iv) the trig function of an angle is the ________________ of the complement.
g) State the 6 Cofunction Identities and state their domains of validity:
© Raelene Dufresne 2011
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Math 150: §5.1 Fundamental Identities
h) Complete the following:
⎛π
⎞
(i) trigθ = _________ ⎜ − θ ⎟
⎝2
⎠
⎛π
⎞
(ii) cotrigθ = _________ ⎜ − θ ⎟
⎝2
⎠
i) What is the co-co-function of a trig function?
j) Why is the term cofunction used for the relationships between sine and cosine, or
tangent and cotangent, or secant and cosecant?
5. Odd-Even Identities (6)
()
a) State the algebraic definition of an even function, f x :
()
b) State the algebraic definition of an odd function, f x :
c) Which trig functions are even?
d) Write the algebraic definitions so that these functions are even:
e) Which trig functions are odd?
f) Write the algebraic definitions so that these functions are odd:
g) Write the 6 Odd-Even Trig Identities and state their domains of validity.
© Raelene Dufresne 2011
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