Pre-Calc Exam Notes 67

Pre-Calc Exam Notes 67 - 2 gives cos 2 sin 2 + sin 2 sin 2...

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Basic Trigonometric Identities Section 3.1 67 Also, from the inequalities 0 sin 2 θ = 1 cos 2 θ 1 and 0 cos 2 θ = 1 sin 2 θ 1, taking square roots gives us the following bounds on sine and cosine: 1 sin θ 1 (3.8) 1 cos θ 1 (3.9) The above inequalities are not identities (since they are not equations), but they provide useful checks on calculations. Recall that we derived those inequalities from the deFnitions of sine and cosine in Section 1.4. In formula (3.3), dividing both sides of the identity by cos 2 θ gives cos 2 θ cos 2 θ + sin 2 θ cos 2 θ = 1 cos 2 θ , so since tan θ = sin θ cos θ and sec θ = 1 cos θ , we get: 1 + tan 2 θ = sec 2 θ (3.10) Likewise, dividing both sides of formula (3.3) by sin
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Unformatted text preview: 2 gives cos 2 sin 2 + sin 2 sin 2 = 1 sin 2 , so since cot = cos sin and csc = 1 sin , we get: cot 2 + 1 = csc 2 (3.11) Example 3.1 Simplify cos 2 tan 2 . Solution: We can use formula (3.1) to simplify: cos 2 tan 2 = cos 2 sin 2 cos 2 = sin 2 Example 3.2 Simplify 5sin 2 + 4cos 2 . Solution: We can use formula (3.5) to simplify: 5sin 2 + 4cos 2 = 5sin 2 + 4 ( 1 sin 2 ) = 5sin 2 + 4 4sin 2 = sin 2 + 4...
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