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Multiple Angle Formulas for Sine and Tangent Big formula day Again, use them, don’t memorize them (6.3, 6.4) (2)

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POD #1 Simplify. Rationalize the denominator.

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POD #2 Use one of yesterday’s formulas to verify the reduction formula. What is the relationship between θ and π/2 – θ ?

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POD #2 Verify the reduction formula. How does this illustrate the relationship between cosθ and sinθ? How about their graphs? How else could we write the cosine side?

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Cofunction formulas Now, we move on to the trig formulas of composite angles of sine and tangent. We will build off of the cosine formulas we’ve already explored. To begin with, however, we need to consider Cofunction formulas; these describe the relationship between the trig functions of complementary angles (get it?)

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Cofunction formulas Sine and cosine, tangent and cotangent, and secant and cosecant are related by their values for an angle θ, and its complement (π/2 – θ).

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Cofunction formulas Although we can prove this in fairly short order, we can also simply look at a right triangle to see why this is so.

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Addition Formula for Sine Let’s derive sin(u + v) using cos(u + v). Use two cofunction formulas, too.

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The Rest We can use this method to derive the rest of the formulas, but we won’t. Instead they are simply presented. You do not need to memorize them. You do need to use them.

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Addition and Subtraction Formulas For sine and tangent.

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Double Angle Formulas For sine and tangent

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Half-Angle Identities/ Formulas For sine and tangent

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So, use them 1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α. Do we care what α is?

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So, use them 1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α. To start with, we’ll need cos α. Think Pythagoras. How many ways could we find that value?

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So, use them 1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α. To start with in one approach, we’ll need cos α. Think Pythagoras. cos α = 3/5

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So, use them 1. If sin α = 4/5, and α is acute, find the exact values of sin 2α and cos 2α. cos α = 3/5 sin 2α = 2(sin α)(cos α) = 2(4/5)(3/5)= 24/25 cos 2α = cos 2 α - sin 2 α = (3/5) 2 – (4/5) 2 = -7/25 How could we check our answers?

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So, use them 2. Verify the identity. Take a deep breath– look at the formulas you have on both handouts. There are two that will work very well.

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So, use them 2. Verify the identity.

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So, use them 2. Verify the identity. Alternate solution.

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So, use it– calculus preview 3. If f(x) = sin x, and h ≠ 0, show

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So, use it– calculus preview 3. If f(x) = sin x, and h ≠ 0, show What is the advantage to having the expression in this form?

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10.3 Double-Angle and Half-Angle Formulas. Half-Angle Formulas After we get the double-angle formula for sine, cosine and tangent, if we make backwards.

10.3 Double-Angle and Half-Angle Formulas. Half-Angle Formulas After we get the double-angle formula for sine, cosine and tangent, if we make backwards.

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