1.6b day 2 Evaluating Trig Functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008

President James A. Garfield National Historic SiteMentor, Ohio

x

y

0

2

3

2

2

First, a little review.

Answer as quickly as you can!

sec 0 1

sin 0

cos 0 1

3cot

2

0

3csc

2

1

3sec

2

undefined

x

y

0

2

3

2

2

First, a little review.

Answer as quickly as you can!

sec2

undefined

cot 0 undefined

sec 1

3sin

2

1

cos2

0

tan 0

We can find exact values for certain angles using special triangles. The first is the 45o triangle.

sin 45

cos45

tan 45 If we superimpose this triangle on a unit circle, the hypotenuse would have a length of 1, and the triangle would look like this:

1

2

1

2

1

1

45

1

12

If we let the two perpendicular sides have a length of 1, then the hypotenuse is: 2 21 1 2c

2

2

2

2

1

x

y

In calculus we frequently do not have to rationalize the denominator, so we can “cheat” and re-label the sides.

0

2

3

2

2

2

2

2

1

4

sin

cos

tan

1 2

22

1 2

22

11

1

x

y

It is not a “unit” circle any more, but these labels let us quickly evaluate the trig functions.

0

2

3

2

1

12

4

sin

cos

tan

1

2

1

2

1

sec

csc

cot

2

1

2

Move the triangle to find other values.

You are welcome to rationalize the denominator any time you like!

x

y

Notice that in this position, the x coordinate is negative.

0

2

3

2

1

12

3

4

sin

cos

tan

1

2

1

2

1

sec

csc

cot

2

1

2

x

y

Here both the x and y coordinates are negative.

0

2

3

2

1

12

5

4

sin

cos

tan

1

2

1

2

1

sec

csc

cot

2

1

2

x

y

In this position, the y coordinate is negative.

0

2

3

2

1

12

7

4

sin

cos

tan

1

2

1

2

1

sec

csc

cot

2

1

2

It is great if you can memorize these or picture the unit circle or special triangles in your mind.

However, a quick sketch and a correct answer is always better than a guess and a wrong answer!