Onward to Section 5.3. We’ll start with two diagrams: What is the relationship between the three angles? What is the relationship between the two chords?

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Presentation transcript:

Onward to Section 5.3

We’ll start with two diagrams: What is the relationship between the three angles? What is the relationship between the two chords? Distance formula:

Square both sides and expand the binomials:

(Note the sign switch in either case.) Cosine of a Sum or Difference

What about the sine of a sum or difference? Start with a cofunction identity: Our new identity!!!

(Note that the sign does not switch in either case.) Sine of a Sum or Difference

Directly from our previous two identities: Tangent of a Sum or Difference Or, use a new formula that is written entirely in terms of tangent:

Use a sum or difference identity to find an exact value of the given expression. Guided Practice 1. Any other way to simplify this expression? Any way to check our work with a calculator?

Use a sum or difference identity to find an exact value of the given expression. Guided Practice 2.

Use a sum or difference identity to find an exact value of the given expression. Guided Practice 3.

Use a sum or difference identity to find an exact value of the given expression. Guided Practice 4.

Use a sum or difference identity to find an exact value of the given expression. Guided Practice 5.

Some problems require us to simply recognize certain trig. identities… Write the given expressions as the sine, cosine, or tangent of an angle. This is the sine of a sum formula!!! This is the cosine of a difference formula!!!

Some problems require us to simply recognize certain trig. identities… Write the given expressions as the sine, cosine, or tangent of an angle. This is the opposite of the cosine of a sum formula!!! This is the tangent of a difference formula!!!

Verifying a Sinusoid Algebraically Previously, we were able to re-write expressions like as Now, we will find an exact value for this function… In general, using the sine of a sum formula: Here, we have: Compare coefficients:

Verifying a Sinusoid Algebraically Solve for a algebraically: Choosing a to be positive gives us: Final Answer: or

Verifying a Sinusoid Algebraically Express the function as a sinusoid in the form Compare coefficients: Solve for a: Verify graphically?

Verifying a Sinusoid Algebraically Express the function as a sinusoid in the form Compare coefficients: Solve for a: Verify graphically?