Ch 7.1: Fundamental Identities and their use In this section, we will 1. revisit, if necessary, previous encountered identities 2. simplify trig expressions 3. verify trig identities, and 4. write a given expression as a trig expression using a suggested substitution. Remark: This section, as well as the subsequent ones in Ch. 7, relies heavily on proofs.
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Ch 7.1: Fundamental Identities and their use
In this section, we will
1. revisit, if necessary, previous encountered identities
2. simplify trig expressions
3. verify trig identities, and
4. write a given expression as a trig expression using a suggestedsubstitution.
Remark: This section, as well as the subsequent ones in Ch. 7,relies heavily on proofs.
Verifying identities
Theorem (Pythagorean identities)
sin2 t + cos2 t = 1
tan2 t + 1 = sec2 t
1 + cot2 t = csc2 t
Why? Let (x , y) be a point on the Unit circle.
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Even/Odd Identities
Theorem (Identities due to symmetry)
sin(−t) = − sin(t), csc(−t) = − csc(t)
cos(−t) = cos(t), sec(−t) = sec(t)
tan(−t) = − tan(t), cot(−t) = − cot(t)
For instance,
1. sin(−π/6) =2. cos(−π/6) =3. tan(−π/6) =
Question) Which of the above functions are odd? even?
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Proof of theorem
Let (x , y) be a point on the Unit circle.
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Example 1Simplify the expression
cos θ + sin θ tan θ.
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Example 2Simplify the expression
cot θ +sin θ
1 + cos θ.
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Example 3Verify that
2 csc2 t =1
1− cos t+
1
1 + cos t.
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Example 4Verify that
tan2 x
1 + sec x=
1− cosx
cos x
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Example 6Write
√4− x2 as a trig expression, using the substitution
sin θ = x2 . Assume 0 ≤ θ ≤ π
2 .
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Homework for Ch 7.1 (pg. 553)
2, 5, 13, 14, 17, 18, 22, 28, 30, 32.Show work to get credit.