Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)

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Presentation transcript:

Inverse Trig Functions redux Some review today, Followed by use and abuse, my favorite (6.6)

SAT #1 Easy

SAT #2 Not quite so easy, but still straightforward

SAT #3 A thinker

Review Inverse Trig Functions What do we already know about the inverse trig functions? List it here.

Review Inverse Trig Functions Among other things: All three inverse trig functions are restricted to half a rotation. Two of the three are continuously increasing. The other continuously decreases. Want a handout?

Review Inverse Trig Functions What do the graphs of the inverse trig functions look like? Sketch them here.

Approaches Either retrieve your chapter 5 handout on inverses or graph on your calculators to fill in these blanks.

Approaches Either retrieve your chapter 5 handout on inverses or graph on your calculators to fill in these blanks.

Key Point A significant characteristic of inverse trig functions is the restriction from the original trig functions. Remember, this is why we often need to build solutions when solving trig equations.

Key Point It can also lead to unexpected answers. Find the following.

Key Point It can also give unexpected answers. Find the following. Why the difference between input and output values? (This is what item 4 on the handout refers to.)

Several problems to work with inverse trig functions Try these.

Several problems to work with inverse trig functions Try these.

Diagrams are useful Rewrite the “arc” part, if it helps, too. Find the exact values.

Diagrams are useful Rewrite the “arc” part, if it helps, too. Find the exact values.

Using multiple angle formulas Find the exact value. Does it matter what the angles are?

Using multiple angle formulas Think in terms of the Subtraction Formula for Sine. We need only sine and cosine values.

Using multiple angle formulas Think in terms of the Subtraction Formula for Sine. We need only sine and cosine values.

Identities Verify the identity. Rather than a totally algebraic verification, see how you can simply reason through it. Consider a diagram.

Identities In other words, show that an angle with a sine of x, and an angle with a cosine of x, are complementary.

Equation with a twist Use inverse trig functions and some fancy algebra to solve this equation.

Equation with a twist Use inverse trig functions and some fancy algebra to solve this equation. Embedded quadratic– is it factorable?

Equation with a twist Use inverse trig functions and some fancy algebra to solve this equation. Embedded quadratic– is it factorable? No. Go to the quadratic formula.

Equation with a twist Substitute, then use the formula.

Equation with a twist Reverse the substitution. Find a single angle.

Equation with a twist Reverse the substitution. Find a single angle. What is the general solution?

Equation with a twist Reverse the substitution. Find a single angle. What is the general solution? ±2πn, ±2πn