SINE! COSINE! TANGENT! INVERSE?!? ... OR An Introduction to the Inverses of the Trigonometric Functions

OUTLINE:

• Definitions of many-to-one and one-to-one functions
• Unit circle
• Sine, Cap-Sine, Arcsin!
• Cosine, Cap-Cosine, Arccos!
• Tangent, Cap-Tangent, Arctan!
• Using Right Triangles

SLIDE 2: BLAST FROM THE PAST! GRADE 12 UNIT CIRCLE UNIT!

• Review of grade 12 precalculus (unit circle unit)
• Ask yourself what angle you're looking for and imagine the unit circle
• We gave the answers to the above questions, but Mr.K then says that one of the solutions in each answer is wrong, but why?
• Introducing the inverses of the trigonometric functions!

SLIDE 3: VOILA! THE UNIT CIRCLE!


SLIDE 4: SINE! CAP SINE! ARCSIN!

• We have the sine function. We want to know its inverse, but we can't since it's a many-to-one function.
• If we restrict the domain of the sine function to [-pi/2, pi/2], we have a function that has the same range as the sine function. This restricted sine function is the Sine (pronounced "Cap-Sine") function.
• Take the inverse of Sine (switch the x- and y-coordinates). We get the arcsine function.
• Note that Sine takes up Quadrants I and IV of sine. Thus, only values of sine found in QI and QIV are in the domain of Sine.

SLIDE 5: COSINE! CAP COSINE! ARCCOS!

• We have the cosine function. We want to know its inverse, but we can't since it's a many-to-one function.
• If we restrict the domain of the cosine function to [0, pi], we have a function that has the same range as the cosine function. This restricted cosine function is the Cosine (pronounced "Cap-Cosine") function.
• Take the inverse of cosine (switch the x- and y-coordinates). We get the arcsine function.
• Note that Cosine takes up Quadrants I and II of cosine. Thus, only values of cosine found in QI and QII are in the domain of Cosine.

SLIDE 6: TANGENT! CAP TAN! ARCTAN!

• We have the tangent function. We want to know its inverse, but we can't since it's a many-to-one function.
• If we restrict the domain of the tangent function to [-pi/2, pi/2], we have a function that has the same range as the tangent function. This restricted cosine function is the Tangent (pronounced "Cap-Tangent") function.
• Take the inverse of tangent (switch the x- and y-coordinates). We get the arctan function.
• Note that Cosine takes up Quadrants I and IV of tangent. Thus, only values of tangent found in QI and QIV are in the domain of Tangent.

SLIDE 7 TO 12: A BARRAGE OF QUESTIONS!

• We found the values of each using the unit circle. Note that some values of the unit circle aren't in the domain of the "Cap functions."

• 3a and 3b are undefined because the angle doesn't live in the domain of Sine and Cos.

• Sine and sine, Cosine and cosine, and Tangent and tangent are inverses of each other. Thus, they undo each other, like multiplying a number by 3 then dividing the number by 3.

SLIDE 13: USING RIGHT TRIANGLES!

• In 8a, we let sin^-1 2/3 = x to make a simplified expression cot x. Looks easier to solve, right? We used the definition of sine (the ratio of opposite side to the hypotenuse side) to determine what cot x is. We solved for cot x.

HOUSEKEEPING:

• Next scribe is Kristina.
• And don't forget to listen to the Dr. Love messages during the last ten minutes of class!